Sean O'Hanlon had to go off injured after just 19 minutes, and he was replaced by Andy Nicholas who has now returned from his loan spell at Chester City.
Grant Smith then fouled Tranmere's Iain Hume on the edge of the area. Ryan Taylor took the free kick and scored to give Tranmere at 20th minute lead.
Christian Roberts should have then equalised for Swindon five minutes later when his shot from inside the area beat the 'keeper but hit the bar, but Swindon were unable to get anything from the follow up.
Fallon also came close to scoring shortly before half time, but his shot was saved, and Tranmere held on to their 1-0 lead going into the half time break.
On 57 minutes Christian Roberts was fouled on the edge of the box, in just about the same spot as Tranmere had scored from in the first half. Grant Smith took the free kick and scored his 9th goal of the season.
Then on 59 minutes Grant Smith made it into double figures when he gave Sindon the lead after a pass from Rory Fallon, Smith was able to score with a powerful straight shot from 10 yards out.
Grant Smith came close a couple more times to getting his first ever hat-trick, but his shots both weny just wide.
Tranmere came close to getting an equaliser, with some of the Tranmere fans starting to cheer as they thought they had scored, but the ball had gone just wide and hit the side netting, and not the back of the net.
Andy Caton came on for the last few minutes to replace Christian Roberts, and Grant Smith once again came close to getting his hat-trick, but Swindon ran out of time to get any more goals as the referee blew up for full time.
Overall a fairly average game, but Swindon deserved their first win for 5 weeks which moves then back up a place to 11th ahaed of next weekend's final away game of the season when they travel to Yorkshire to play Huddersfield time.