The former Wigan and Bradford hitman found the back of the net with a sweet 37th minute curling strike, before bagging a second midway through the second half as Brian Carey's struggling side battled bravely for full points.
And victory was extra special having been thumped 4-0 at home to Chesterfield on Tuesday night.
Accrington, buoyed by their 5-0 thrashing of the Dragons in this fixture last season - one in which they scored five times in 26 minutes - were the better starters.
Peter Cavanagh had a good early chance, but his powerful shot was well-saved by Wrexham keeper Anthony Williams.
The visitors nearly scored themselves three minutes later when Michael Proctor fired over the bar with a rasping 25-yard drive.
Danny Williams forced Accrington's Ian Dunbavin to make a smart stop a couple of minutes later, before Chris Llewellyn saw a solid saved as well by the on-song keeper.
Paul Mullin headed a Neil Taylor free-kick wide of the target as the home side looked for the opener, but John Coleman's side were hit by a sucker-punch.
Roberts was cool under pressure and beat Dunbavin with a precision strike.
Andy Proctor should have equalised for Stanley four minutes before the break, but he headed wide from close range.
Coleman's charges were again the better starters in the second half and in the 56th minute David Brown shot wide.
Steve Evans had a header saved by Dunbavin as the visitors looked to put the game out of reach, while at the other end Brown went close again, this time though his shot was saved by Williams.
Then it was Roberts who turned the tide in his team's favour with a clinical finish in the 66th minute.
Behind to a second goal, Stanley chased the game and Roscoe D'Sane went agonisingly close, but his goalbound effort was saved by the impressive Williams.
Lee McEvilly forced Williams to save with quarter of an hour to go and a minute later headed wide.
But no matter how hard the home side tried, they just couldn't score and in the 79th minute were lucky not to go 3-0 down when Evans had a shot saved by Dunbavin.