Conlon, released by the Stags earlier in the week, headed a vital breakthrough six minutes into first half injury-time - only Mansfield's second goal in six games.
And he then set-up a second for Simon Brown in the 66th minute to clinch the Stags' first win in nine matches and avoid a club record fifth successive home defeat.
The Bees, fresh from a five-goal mauling at Grimsby and without a win in three games, staged a late fight back when former Mansfield striker Adam Birchall tapped in a minute later.
But two dramatic saves from home keeper Jason White preserved the Stags' lead as they held on for the vital victory.
The Stags, desperately fighting a crippling injury list which robbed them of six first teamers, were given a massive boost just before kick-off when influential midfielder Stephen Dawson made a surprise return after a late fitness test.
But their search for the elusive point to banish any relegation worries began badly with passes going astray and few chances created in a nervous opening half devoid of any home goalmouth action until six minutes into first half added-on time.
At the other end Jason Puncheon almost gave Barnet the perfect start, his low 20-yard drive kicked away by keeper White after a flowing cross-field move also involving Liam Hatch.
Nick Bailey's speculative, dipping 40-yard shot almost caught out keeper White as Barnet continued to look the better side.
However, the Stags grabbed their precious lead in first half added on time. Barnet keeper Lee Harrison beat away Conlon's close-range header from a deep Brown cross, but the rebound was bundled in by the striker in his last home game for the club.
Nine minutes into the second half Puncheon again went close for Barnet, hooking his drive through a crowded goalmouth and narrowly wide.
The Stags doubled their lead in the 66th minute when Conlon's clever touch put Brown clear of the defence and he coolly rolled the ball past keeper Harrison.
That should have secured their first win in nine games, and their Football League status, but within 60 seconds Barnet grabbed a lifeline. Barry Cogan and Dean Sinclair both hit the post in a goalmouth scramble before ex-Stag Birchall tapped in from close range.
Two minutes later only a fine save by White denied Bailey an equaliser as the nervy Stags started to panic. Alex Baptiste's poor clearance fell straight to Bailey and his first-time shot had the keeper scrambling across his line to save.
Danny Sleath's acrobatic 75th minute drive forced Harrison into a flying fingertip save as the Stags looked to make the points safe, and that stop almost proved vital in injury time when Barnet substitute Oliver Allen stretched to head wide with home keeper White beaten.