Christian Roberts struck twice in the final 20 minutes to put League Two Swindon through to the second round at the expense of League One Carlisle.
The Robins took the lead when Chris Lumsdon put a header into his own net, before Carlisle equalised when Kevin Gray nodded in from Lumsdon's corner.
Roberts scored his first when he drilled home from 16 yards after being let through on goal by Gray.
He added his second from 12 yards after converting Lucas Jutkiewicz's ball.
606 DEBATE: Give your thoughts on the match
Swindon: Phil Smith, Jack Smith, Nicholas, Ifil, Williams, Shakes, Evans, Weston, Brown (Zaaboub 41), Roberts, Brownlie (Jutkiewicz 90).
Subs Not Used: Stewart, Pook, Caton.
Booked: Zaaboub, Evans.
Goals: Lumsdon 9 og, Roberts 70, 90.
Carlisle: Westwood, Raven (Livesey 46), Aranalde, Murphy, Gray, Harper, Lumsdon, Thirlwell, Gall, Holmes (Hindmarch 81), Hawley.
Subs Not Used: Williams, Billy, Grand.
Booked: Lumsdon, Aranalde, Thirlwell.
Goals: Gray 43.
Att: 4,938.
Ref: K Wright (Cambridgeshire).