The league leaders looked well up against at the halfway stage of the contest, needing 292 for victory after Jordan King smashed a sensational 151 from just 138 deliveries.
The Darley opener shared a mammoth opening-wicket stand of 206 with Adam Copeland (55) and appeared to have put the game out of Birstwith’s reach.
But the hosts were not to be denied and James Riley (79), Tom Croston (72), and Callum Halliday (45 not out) led the way over the line with 10 balls to spare.
Saturday’s result moves Birstwith 31 points clear of Darley at the summit, albeit having played a match more.
Goldsborough are up to third courtesy of a seven-wicket success over rock-bottom Burton Leonard.
Max Crompton hit a fine 70 to steer the strugglers to 179, Mark Wood(5-41) excelling with the ball in hand for the home side.
Damon Ive (106 not out) then showed all his experience, scoring a superb century alongside Jarrod McPhee to wrap up the spoils.
Fourth-placed Masham extended their fine run of recent form by beating Pateley Bridge.
The Badgers won the toss and elected to bat first, but might have been regretting that decision when they slumped to 79/7, Craig Broadley claiming 4-41.
Joe Preece rescued his side with a knock of 67, but his efforts proved to be in vain as Tom Smith (49) and Andrew Sturdy (44 not out) saw Masham to victory with four wickets in hand.
Blubberhouses made it three wins on the spin when they crushed Kirk Deighton, Mark McEneaney (122) and Luke Davis (72) going big in a huge first-innings score of 362.
Three scalps apiece for Sajid Khan, Adil Shazad and McEneaney then got rid of Deighton on 130, Mike Malthouse striking 61 not out.
Second-from-bottom Ouseburn remain deep in trouble having failed to overhaul Harrogate 3rds’ 217/9.
Will Jenkins’ 35 was the best the hosts could muster as Ollie Bareham helped himself to four wickets.
Earlier, Brian Fenwick (52) and Charlie Holtham had impressed with the bat for the Roosters in the face of some excellent bowling by Jonty Moorhouse (5-49).
West Tanfield’s Angus Shaw (6-35) and Joshua Gericke (4-30) ripped through Helperby’s batting line-up, dismissing the visitors for 120 and setting up a seven-wicket triumph.