But with four games to go they still can't shake off defending champions Beckwithshaw, who they meet in the penultimate week.
Beckwithshaw faced a difficult fixture at fourth-placed Collingham but spinner Richard Pearson (6-37) restricted the home side to 165-9 and John Inglis (88) helped seal the four-wicket win.
Nick Brook (5-37) and Jason Wright (4-24) skittled Rawdon for 63 before Michael Truswell (41) led Burley's victory chase that leaves them seven points clear.
Rawdon are still too close to the drop zone for comfort but will be relieved that the three sides below them, all of whom they meet in the run-in, also lost.
Kirkstall Educational's Jos Siddall claimed the last wicket with four balls to spare to leave North Leeds adrift at the bottom.
Adam Waite hit 92 not out in Guiseley's 183-5 before Abid Awan (6-41) dismissed Adel for 115 and a century from Sam Smith and 52 not out from Josh Atkinson clinched an eight-wicket win for Otley at Pool.
Ishtikhar Hussain (6-46) and Michael Abbott (60no) were the stars of Green Lane's win at Thackley but they remain 19 points behind the leaders with games running out.
Division Two leaders Bilton crashed to a shock defeat at Knaresborough and, having taken only one point from the last 18, they have seen their lead slashed to five points. Elias Karageorgiou hit 62 before Jonathan Corcoran (29no) and Peter Barnes, who followed up a 3-17 spell with 22 not out, clinched the win.
Of the chasing pack, only Follifoot took advantage, James Pickles (78no) starring in a six-wicket win over Calverley for whom Shahid Bashir hit 64 not out.
That took them above Harden who fell victim to Tong Park Esholt veteran Mick Smith (6-67), while a 6-26 spell from Tom Baines and unbeaten half-century from Cameron Taylor gave Upper Wharfedale victory over fourth-placed Olicanian.
Gary Boulton (5-29) and Jonathan Best (80no) starred as Steeton eased past Illingworth with nine wickets in hand while a fine all-round display from Luke Taylor, who followed up a half century with 3-54, helped Silsden take four points at Bolton Villas.