The Chairboys were dominant from start to finish and fully deserved their win through goals by Ian Stonebridge, Nathan Tyson and Jonny Dixon.
The visitors almost took the lead within the first 90 seconds when Swindon failed to cut out Stefan Oakes' corner and Roger Johnson saw his header cleared off the line by Charlie Comyn-Platt.
Oakes saw a 30-yard thunderbolt flash just wide of the post moments later before Wycombe took a deserved lead on 12 minutes.
Veteran midfielder Rob Lee played in Stonebridge down the left and he fired in an unstoppable left-foot shot past Robins keeper Tom Heaton.
But Wycombe had keeper Frank Talia to thank on 23 minutes when he reacted quickly to a deflection from Gareth Whalley's left-foot shot to palm the ball around the post.
And the former Swindon keeper was again called into action twice in five minutes when he denied Town midfielder Colin Heath.
Heath's first effort was well-saved by Talia but seemed to be heading goalwards until defender Michael Williamson was on hand to clear.
And Talia was again alert moments later when he pushed away Heath's fierce volley from the edge of the box.
But Talia had no chance 12 minutes after the break when Michael Pook cut in from the left to fire in an unstoppable shot from the edge of the area.
But Wycombe reacted well to conceding a goal and Johnson saw a header come back off the post before they regained their lead on 62 minutes.
Swindon failed to cut out Talia's long kick and Tyson ran free to fire home. Wycombe wrapped up the win on 74 minutes when Clint Easton's free-kick was headed onto the post by Williamson, leaving substitute Dixon the easy chance of tapping in the rebound.
Swindon substitute Jamie Cureton flashed a late shot high and wide as Wycombe held on to secure their passage through to the next round.