However, two tries at the death put a lopsided look on the scoreline and denied the visitors a point their graft had deserved.
Not for the first away trip this season, or even this month, West were forced into some 11th hour changes.
Colin Cheslin and Stephen Stockdale were both withdrawals, with Zac Southern and Ben Lloyd coming into the starting XV on the left wing and lock. Both players put in tremendous shifts.
It was pretty much all Birkenhead Park in the opening 15 minutes as the hosts looked to atone for the Merseyside derby drubbing at Waterloo seven days earlier.
They looked anxious though so when they tried to move the ball wide Greg Edwards, fast becoming a modern-day Mike 'Burglar Bill' Harrison made the interception.
He only reached halfway but the centre had saved West from going 7-0 down.
With Dave Hall missing an early penalty that had three points written all over it, it was Birkenhead Park feeling the frustration, especially when the full-back spilled an 18th-minute kick ahead by Stuart Waites.
Waites couldn't believe his good fortune and he ran clear for a try. For good measure he put over the conversion from wide on the right.
West's defence was worked hard but they put their bodies on the line.
And they were unlucky to concede a 'try' just after the half-hour.
Birkenhead had been held up over the line and gained a five-metre scrum. The hosts chipped away and hooker and skipper Christian Jones drove over in the right corner.
Mr Ramsden could not see it and after cries of "yes" from home players and fans, he ran around to the blindide and, incredibly, awarded the score.
Jones may have got the ball down, but the Yorkshire ref hadn't seen it and should not have guessed.
To rub salt in the wounds, Hall converted from the right touchline.
Mr Ramsden added over six minutes of stoppage time and right in the final seconds penalised West for being offside in front of their posts.
Thankfully for the visitors Hall was not as accurate in front of the sticks as he was from out on the flanks and half-time arrived with the scores level at 7-7, even if the Birkenhead scoreboard read 7-5 to the hosts.
Waites had no such problems 10 minutes into the second half when he put West 10-7 ahead after their best passage of play.
Frustration reared its head again when Jonny Boatman was almost decapitated by a head-high tackle at the re-start.
The Birkenhead prop escaped with a ticking off but when Gareth Foreman stuck his hand in a ruck five minutes later the yellow card came out of the referee's shorts pocket.
Down to 14 men, West held out until the 62nd minute when they got winger Mike Chapman over in the left corner after West ran out of defenders. Hall added a super conversion.
Five minutes later, West frustration boiled over when they were penalised just inside the home half.
Dissent meant they were marched back though for what seemed to be further than the mandatory 10 metres.
It brought the award to within kicking distance and Hall was on target from 39 metres.
West did get some rub of the green when Mr Ramsden missed a knock-on by Andrew Davies.
The lock momentarily stopped before putting his foot on the accelerator and powered up the left wing.
Reminisent of a try at Alnwick a season or four ago, Davies looked like going all the way before being forced out on the 22.
Frustration again at the resultant line-out. Mr Ramsden penalised West and it was the last time they caught sight of the home line.
West conceded several penalties late on and when Park kicked one award into the away 22, the forwards got a drive on and sub Olly Mabbutt finished it off with an unconverted try in the final minute.
It took a bonus point away from West and in injury time, Birkenhead returned to the 22 where Jones forced his way over for his second try and a BP for his team.
Frustration was the only word for it.
Birkenhead Park: Hall, Houlistan, Hughes, Scott, Chapman, Rich, Lowry, Cassidy, Jones, Campbell, Saidi, Stead, Ashcroft, Stead, D Ibbotson. Reps: Mabbutt, M Ibbotson, Ingram
Tries: Jones [2], Chapman, Mabbutt. Con: Hall [3]. Pen: Hall
West: Waites [Mallinson 72], Butcher, Edwards, Robinson, Southern, G Foreman, A Foreman, Wood, Bailey [Hogan 55], Coates, Davies, Lloyd [Eddy 50], D Boatman, J Boatman, Taylor
Try: Waites. Con: Waites. Pen: Waites
Referee Nick Ramsden [Yorkshire Federation]
Not for the first away trip this season, or even this month, West were forced into some 11th hour changes.
Colin Cheslin and Stephen Stockdale were both withdrawals, with Zac Southern and Ben Lloyd coming into the starting XV on the left wing and lock. Both players put in tremendous shifts.
It was pretty much all Birkenhead Park in the opening 15 minutes as the hosts looked to atone for the Merseyside derby drubbing at Waterloo seven days earlier.
They looked anxious though so when they tried to move the ball wide Greg Edwards, fast becoming a modern-day Mike 'Burglar Bill' Harrison made the interception.
He only reached halfway but the centre had saved West from going 7-0 down.
With Dave Hall missing an early penalty that had three points written all over it, it was Birkenhead Park feeling the frustration, especially when the full-back spilled an 18th-minute kick ahead by Stuart Waites.
Waites couldn't believe his good fortune and he ran clear for a try. For good measure he put over the conversion from wide on the right.
West's defence was worked hard but they put their bodies on the line.
And they were unlucky to concede a 'try' just after the half-hour.
Birkenhead had been held up over the line and gained a five-metre scrum. The hosts chipped away and hooker and skipper Christian Jones drove over in the right corner.
Mr Ramsden could not see it and after cries of "yes" from home players and fans, he ran around to the blindide and, incredibly, awarded the score.
Jones may have got the ball down, but the Yorkshire ref hadn't seen it and should not have guessed.
To rub salt in the wounds, Hall converted from the right touchline.
Mr Ramsden added over six minutes of stoppage time and right in the final seconds penalised West for being offside in front of their posts.
Thankfully for the visitors Hall was not as accurate in front of the sticks as he was from out on the flanks and half-time arrived with the scores level at 7-7, even if the Birkenhead scoreboard read 7-5 to the hosts.
Waites had no such problems 10 minutes into the second half when he put West 10-7 ahead after their best passage of play.
Frustration reared its head again when Jonny Boatman was almost decapitated by a head-high tackle at the re-start.
The Birkenhead prop escaped with a ticking off but when Gareth Foreman stuck his hand in a ruck five minutes later the yellow card came out of the referee's shorts pocket.
Down to 14 men, West held out until the 62nd minute when they got winger Mike Chapman over in the left corner after West ran out of defenders. Hall added a super conversion.
Five minutes later, West frustration boiled over when they were penalised just inside the home half.
Dissent meant they were marched back though for what seemed to be further than the mandatory 10 metres.
It brought the award to within kicking distance and Hall was on target from 39 metres.
West did get some rub of the green when Mr Ramsden missed a knock-on by Andrew Davies.
The lock momentarily stopped before putting his foot on the accelerator and powered up the left wing.
Reminisent of a try at Alnwick a season or four ago, Davies looked like going all the way before being forced out on the 22.
Frustration again at the resultant line-out. Mr Ramsden penalised West and it was the last time they caught sight of the home line.
West conceded several penalties late on and when Park kicked one award into the away 22, the forwards got a drive on and sub Olly Mabbutt finished it off with an unconverted try in the final minute.
It took a bonus point away from West and in injury time, Birkenhead returned to the 22 where Jones forced his way over for his second try and a BP for his team.
Frustration was the only word for it.
Birkenhead Park: Hall, Houlistan, Hughes, Scott, Chapman, Rich, Lowry, Cassidy, Jones, Campbell, Saidi, Stead, Ashcroft, Stead, D Ibbotson. Reps: Mabbutt, M Ibbotson, Ingram
Tries: Jones [2], Chapman, Mabbutt. Con: Hall [3]. Pen: Hall
West: Waites [Mallinson 72], Butcher, Edwards, Robinson, Southern, G Foreman, A Foreman, Wood, Bailey [Hogan 55], Coates, Davies, Lloyd [Eddy 50], D Boatman, J Boatman, Taylor
Try: Waites. Con: Waites. Pen: Waites
Referee Nick Ramsden [Yorkshire Federation]