It was a third win in succession, with George Moncur and Callum Harriott getting the goals to extend.
Moncur's came as early as the third minute and Harriott made it 2-1 after thomas had equalised.
The win took the U's up to tenth in the League One table, ahead of Tuesday evening's game against Bradford City.
The U’s were on the offensive from the first whistle and left back Matt Briggs had already got forward a couple of times before he created the opening goal in the second minute.
He flicked the ball forward to Callum Harriott, picked up the return and pulled the ball back for George Moncur to sweep home past the home keeper Tyrell Belford.
That early goal spurred the Robins forward and Jamie Jones made a save at the near post to thwart Thomas, before launching the U’s on a n attack of their own.
Harriott picked up his keeper’s long throw, drove forward but was denied by a save from Belford.
The home side had a great chance to equalise on ten minutes, but Thomas shot over from ten yards when he really should have done much better.
He did do better just over ten minutes later, when he found himself in space as a corner was flicked on and volleyed home giving Jones no chance.
Chances were then exchanged, with Owen Garvan for the U’s and Ojama for the hosts failing to beat their opposing keepers.
George Moncur and Joe Edwards just missed the target with efforts as the U’s got forward more, but their respective shots went wide and over.
Marvin Sordell was getting on the ball in dangerous areas but couldn’t create an opening for himself, but he helped create another chance on thirty six minutes.
He was fouled but the referee played a good advantage and Harriott was denied by a relatively straight forward save by Belford.
Massey was next in the action, combining with Olufemi on the right, before drilling a shot goalwards that was blocked before it reached the home goal.
The U’s pressure told four minutes before half time, as they sliced through the Swindon defence to make it 2-1.
Garvan to Harriott, back to Garvan and the classy midfielder then found Sordell just inside the home penalty area.
Harriott had continued his run and Sordell spotted it and deftly found his onrushing teammate.
Harriott took a touch to steady himself, before guiding his shot past Belford and into the back of the net.
On half time, the U’s had one final shot, Briggs effort saved, but the U’s were good value for their half time lead.
After two successive defeats, the home side would have been wanting a big reaction in the second half and they started on the front foot.
Matt Briggs slipped as he defended Kasim but Tom Eastman was on hand at the near post to block the danger.
The U’s were playing more on the break now but they almost extended their lead on fifty eight minutes through Owen Garvan.
Moncur hit a delightful through ball into Sordell’s path and the U’s frontman was clear down the left.
He picked out Garvan on the edge of the box but the shot hit a defender and went out for a corner, which came to nothing.
Jamie Jones made a diving save to deny Branco scoring with a header from a corner as the home side continued to dominate second half possession.
That wasn’t to say that the U’s chances had dried up, Moncur had a shot saved comfortably by Belford before the home keeper kept his side in it with a fantastic tip over from Massey.
Harriott was replaced by Darren Ambrose before the corner could be taken and from the resultant set piece, Frankie Kent headed just past the post.
The home side missed a glorious chance to equalise on seventy six minutes, with Ajose the guilty man.
A cross from the right found its way to the far post and although he was stretching, the Swindon forward hit the post when he should have scored.
With twelve minutes to go, Alex Gilbey came on for Moncur and much of his work was done helping the U’s defend their one goal lead.
With three minutes to go, the third change saw Macauley Bonne came on for Sordell and he managed to kill time on a couple of occasions chasing long balls deep into the Swindon half.
With a minute to go, Branco won another header from a corner but Tosin Olufemi, playing his first game for over a year, headed off the line.
Jones had to touch a deflected shot wide in injury time but that was the final chance and the whistle went seconds later.
It was the U’s third win in succession and lifted them to tenth place in the League One table.