It was only the second time that Paul Ince's outfit had lost away all season since they went down at Rochdale back in September.
The game followed a corner which the visitors' defence failed to clear midway through the second half. Chris Llewellyn placed his flag kick into the area where defender Steve Evans went up for a header that was weakly punched away by Willy Gueret.
It landed 20 yards out for the Wrexham skipper to score his superb winner. Spotting Gueret off the line, Roberts produced a perfect chip that cleared the bunch of defenders and dipped under the bar.
It was no less than Wrexham deserved for a spirited performance by a team that included half a dozen new faces brought in by manager Brian Little in the transfer window, as he starts to rebuild the side.
Conditions were poor after heavy rain, causing comment from Ince, who complained after finding he was unable to bounce a ball on the sticky surface before the game.
He later said that conditions suited Wrexham and stopped the Dons from playing their normal game.
There were few goalmouth incidents with neither side looking likely to score until Roberts broke the deadlock.
Colin Cameron headed wide from an early Dons chance and later Jemal Johnson saw a shot blocked. From another opening set up by a Dean Lewington corner, Drissa Diallo headed narrowly wide.
Llewellyn had a shot held low down by the Dons keeper and Evans headed across goal from close in.
Llewellyn also had another shot blocked and Evans saw a second header clearing the crossbar.