Pete Hardisty's team were pipped to top spot by the men from Crayke Lane last term, but landed an early blow in this season's title fight courtesy of Saturday's one-wicket success in what was a predictably close contest.
Larry Audain (80) was the only Darley player to make any real impression with the bat as the hosts posted 181/8 from their 45 overs, Sam Barker taking 3-38.
An unbeaten half-century from Craig Robinson (67 not out) alongside James Riley (43) then saw Birstwith edge over the line with nine deliveries to spare despite the best efforts of Kabir Bashir, who bowled superbly and returned 6-20.
Newly-promoted Arthington also won by a single-wicket margin, making a fine start to life in the top-flight by beating Masham on the road.
Rahul Khode bagged 4-17 to help restrict the home side to 144/9, Sam Ambler hitting 35.
Having reached 76/2 in reply, Arthington looked to be on course for a routine win, but the dismissal of skipper Naveed Andrabi (31) sparked something of a collapse.
Harry Hill helped himself to four scalps, though the visitors eventually reached 145/9 in the 41st over.
Runners-up in the second tier last year, Killinghall also got off to a flyer in Division One as they sank Goldsborough.
A trio of wickets apiece for Luke Haidar and Hamish McIntyre ensured that the away team progressed no further than 118, Mark Wood contributing 33 of that total.
Wicket-keeper Dave Wallis then struck an unbeaten 43 to guide Killinghall past their target.
Pateley Bridge just about overcame Blubberhouses in a thrilling, high-scoring affair which went right down to the wire.
Badgers opener Mason Scopelliti laid the foundations for a successful run chase, smashing 105 from just 111 balls as the hosts made their way to 279/7 with two balls of the match left to be bowled.
Earlier, Lee Platts (86), Will Haines (75) and Barrie Fewster (43) had all weighed in to guide the visitors to 276 all out, Joe Preece snapping up four wickets.
A superb knock of 95 not out from Henry Blythe enabled Kirk Deighton to get the better of Helperby on home soil.
He was the only batsman to show any kind of form with the willow in hand in a total of 147/9 as Dan Marston bagged 4-50.
It was a similar story for Helperby, with only Tom Messenger (77) impressing in their 143/9, Mark Harland and Thomas Conboy both effecting a trio of dismissals.
Harrogate 3rds endured an afternoon to forget as they were sent packing with just 42 runs on the board at West Tanfield.
Jake Stubbs (5-14) and Jonny Luty (4-17) did the damage for the home team after Harry Lamb (33) had top-scored in their innings of 155 all out.
Rob Stanworth was the pick of the Roosters' attack, claiming 4-13.