We actually had a chance to win when Eldon Myers chased down his kick through for a try which would have given him a deserved hat trick.
But his luck was out as the ball rolled agonizingly over the dead ball line.
It fired up the home side who attacked the try line with referee Sam Phillips eventually losing patience with West infringements to march between the posts with no time left for the visitors to mount a response.
Wet began the game with five players missing after last week's epic victory against second placed Morley.
Playing up the slope and into the strong wind we played some excellent possession rugby and dominated the first half scrums
However, it was the home side who went ahead through young fly half Will Brigham with the conversion rebounding off the post.
We continued to dominate the set piece but fell further behind five minutes later when Adam Coates dropped the ball close to the half way line and home centre Chris Saddler gratefully picked up to race home to give his side a 10-0 lead.
Wet came back with our pick and drive forcing the hosts back and the pack got their reward after a scrum close to the home line which saw John Bunter touchdown, with Zac Southern's conversion narrowly wide.
It looked like that would be the last score of the first half with West looking forward to having the elements in their favour for the following forty minutes.
Unfortunately, tries either side of the break thwarted those plans.
The first came deep into an added six minutes when a home free kick went to Saddler who somehow managed to slide his way through some poor tackles to give his side a 15-5 lead at the break.
That was swiftly increased from the restart when our kick off failed to go ten metres. Driffield attacked from the resulting scrum with burly prop Felix Nellon bustling over.
We were facing a 5 point deficit but set up about the task with Zac Southern managing one of the few penalty attempts of the afternoon before Eldon Myers got in on the act.
His first try came from a break from midfield which saw him gallop in under the posts to set up an easy conversion - his second giving the West fly half a much harder attempt from wide out which was missed.
However, Southern did manage a penalty with five minutes left on the clock which levelled the scores at 20-20.
We were down to 14 men at this point, after replacement Rob Thorn was sin-binned as we defended on the line, and he was closely followed by skipper Dan Boatman as Driffield desperately sought their winner. Outnumbered, we finally capitulated with that penalty try deep into injury time.
Driffield: Alex Piercy, Sam Furbank, Ed Askew, Chris Saddler, Gary Stephenson, Will Brigham, Simon Ord, Felix Nellon, Guy Brumfield, Paul Dervey 9captain), Oliver Borman, Jesse Harris, Sam Havler, Adam Brankley, Henry Forbes.
Reps (all used); Olly Newborn, Simon Pettinger, Jeremy Harrison.
Tries: Brigham, Daddler (2), Nellon, en try.
Cons: Brigham (2)
West: Luke Mallinson, Eldon Myers, Peter Youll, Greg Edwards, Zac Southern, Ryan Painter, Adam Coates, Andrew Rollins, Michael Hogan, Owen Bennett, John Bunter, Jay Corbett, Dan Boatman (captain), Sam Miller.
Reps (all used): Rob Thorn, Jack Angus, Alex Barker.
Tries: Bunter, Myers (2).
Cons: Southern (1)
Pens: Southern (1)
Yellow cards: Thorn, Boatman.
Referee: Sam Phillips (London)