Oates won the game with 17 minutes to go, registering his fourth goal of the season in fine style.
Ahead of the game, Stags boss Nigel Clough opted to make four changes to his starting line-up following the defeat at Sutton last time out - a result which had following five straight wins in all competitions.
George Maris, Jordan Bowery, Harry Charsley and Ollie Clarke were all dropped to the bench for long trip to West Sussex, with George Lapslie, Will Forrester, Farren Rawson and Oates - who had been out for a fortnight - all earning recalls.
Once the action got under way, the Stags produced a hugely competent first-half performance and deservedly led at the interval.
Oli Hawkins fired the visitors ahead in the 24th minutes when he bagged his fifth goal of the season.
Ryan Stirk clipped in a precise cross for Hawkins, and he volleyed home superbly.
In bitterly cold conditions, the Stags created an early opening when Stirk fired an opportunity narrowly off target.
At the other end, Ashley Nadesan ought to have done better with his opportunity from inside the Stags’ box.
After breaking the deadlock, Hawkins was presented with another great chance by Stephen McLaughlin, but he failed to find a finish.
Crawley, who hadn’t won on their own patch since September, bagged a leveller three minutes after the restart.
James Tilley provided the brilliant finish, lashing home into the top corner from outside the box.
The hosts’ tails were up, and just past the hour mark the woodwork saved Mansfield.
This time Jake Hessenthaler fired in an angled strike which struck the crossbar before fizzing to safety.
The Stags responded positively, however, and two minutes after that full-back Elliott Hewitt latched on to Lapslie’s pass before firing an effort narrowly off target.
Oates then made no mistake when he slotted home smartly after meeting Hawkins’ measured pass.
After Stags’ sub Tyrese Sinclair hit the post in the 91st minute, the hosts went in search of a second equaliser even later, and Kwesi Appiah headed against the crossbar.
However, the Stags’ rearguard had done enough following a frantic finish.
Crawley: Morris, Francomb, Francilette, Craig, Davies, Tilley (Lynch, 90), Hessenthaler, Powell, Frost (Nichols, 46), Appiah, Nadesan (Grego-Cox, 79).
Unused subs: Noukeu, Kowalczyk, Bansal-McNulty, Marshall.
Stags: Bishop, Hewitt, Rawson, Forrester, McLaughlin, Stirk, O’Toole, Lapslie (O Clarke, 90), Oates (Sinclair, 79), Quinn (Maris, 69), Hawkins.
Unused subs: Shelvey, Charsley, Bowery, Burke.
Referee: Andy Davies