Jutkiewicz equalised for Swindon after Richard Walker had opened the scoring for Rovers, before Weston put his side in the lead just before the break with a stunning 30-yard strike to secure all three points for Swindon.
Bristol Rovers had the first opening of the match when Walker picked up a Craig Hinton pass but his shot went well wide.
Former Aston Villa trainee Walker then opened the scoring on 12 minutes.
Rovers' goalkeeper Steve Phillips punted the ball downfield and the Swindon defence went walkabout. Walker picked up the ball and slotted past Phil Smith from 12 yards.
However 17-year old Jutkiewicz equalised for Swindon on 15 minutes. He shot right-footed past Phillips in the Rovers goal after picking up a long ball.
Weston gave his side the lead at the end of a frenetic five-minute spell. The midfielder struck a right-footed shot from 30 yards that hit the bar and bounced over the line and out again.
Midfielder Aaron Brown headed the ball into the net, but the linesman had already given the goal.
Walker hit back for his side with a right-footed shot that went over the bar.
Rovers' midfielder Craig Disley should have equalised just before half-time but his 16-yard shot went wide of the right-hand post.
In the second half Walker linked up with Stuart Nicholson and Nicholson shot straight at Smith in the Swindon goal with the net at his mercy.
Walker was proving a constant thorn in the Swindon side and he tried his luck with a right-footed shot from 25 yards on 79 minutes that Smith did well to keep hold of.
Swindon should have extended their lead when Jutkiewicz got on the end of a Brown pull-back and managed to shoot wide from six yards out.
Bristol Rovers substitute Andy Sandell had a fantastic chance to equalise deep into injury time. Sandell met a free kick with a free header from 15 yards but Smith dived full length to save.