Lincoln took the lead on 13 minutes with a close range shot by Junior Mendes after he had been left unmarked in the box.
Swindon equalised on 17 minutes when Jutkiewicz crossed the ball to Roberts who scored from 6 yards out to make it 1-1.
New Signing Barry Corr put Swindon into the lead on 35 minutes with a close range making it 2-1 to Swindon which they held onto going into the half time break.
Lee Peacock made it 3-1 to Swindon with a shot from 6 yards out just after the break on 49 minutes.
Swindon came close a few more times including a long range shot which struck the bar and went over.
Stalland got a second goal for Lincoln on 77 minutes again from close range.
Lincoln pushed forward to get an equliser and came close a number of times but despite 5 minutes of stoppage time ran out of time and Swindon picked up their first away win for 2 months.
Overall a good game with plenty of chances from both sides, but Swindon for once took their chances well and deserved the win which moves them back up to 4th place ahead of next Saturday's game against Shrewsbury.