Ben Holderness' men were beaten by 41 runs when they visited fellow strugglers New Rover on Saturday afternoon, but the eight points that they finished the contest with proved sufficient to move them up one place in the top-flight standings.
Oliver Hebblethwaite (3-42) and Joe Holderness (3-56) performed well enough with ball in hand as the hosts were dismissed on 237 from the final ball of their 50 overs.
Hebblethwaite then completed an excellent afternoon's wok by hitting 71 from just 69 balls when 'Shaw replied, Ross Sedgley (70) also impressing with a half-century of his own.
And although Pete Hotchkiss added 33, the Stags failed to progress any further than 196.
Kayne Ramsay impresses on Harrogate Town debut just hours after meeting team-mates for first time following Southampton FC switch
Harrogate Town 0 Crawley Town 0: Advertiser player ratings
Harrogate Town 0 Crawley Town 0: Sulphurites 'weren't sharp enough in final third'
Kayne Ramsay signs for Harrogate Town from Premier League Southampton FC
Harrogate Town v Crawley Town: Sulphurites aiming to get back on track against toothless Red Devils
Beckwithshaw's escape from the bottom two came at the expense of local rivals Bilton, who were defeated for the fifth time in as many matches.
Beaten in agonising fashion last time out, David Cummings' team came out on the wrong side of another close contest having put 179 on the board away at North Leeds.
Tuseif Arshad (40) and Jacques Allen (31) contributed useful runs, however no visiting player was able to go big.
Some tight Bilton bowling and fielding ensured that the game remained in the balance right up until the latter overs, though North Leeds eventually made it over the line with eight balls to spare.
Collingham & Linton stay fifth in the table despite being blown away for just 77 by Horsforth Hall Park.
The away side looked to be in a decent position at the halfway stage of the contest having restricted their hosts to 161 all out.
Nadeem Shefta did most of the damage, serving up a near-unplayable spell of bowling on his way to recording superb figures of 6-18.
Toby Jacklin also did his bit, claiming 3-45, but Collingham's batsmen failed to fire, with none of them managing more than skipper Jonathan Haslem's score of 19.
Captain Nick Robinson was left stranded one run short of his century as Follifoot got their Division Two promotion push back on track following consecutive defeats.
His knock of 99 not out against Harden was scored from only 65 balls and came after James Orton had hit 91, while Yasar Ali (57) and Tom Stewart (48) also shone in a mammoth first-innings total of 328/4 from 50 overs.
Elliot Jennings and Ali then bagged a trio of scalps apiece as the visitors were bowled out for 256 in reply, the result lifting Follifoot back up into second place.
Henry Wainman bagged six wickets for Bardsey, but they were then bowled out for 132 by Horsforth chasing 186.