Substitute Adam Birchall scored the equaliser to earn Barnet a replay.
Swindon's Billy Paynter headed wide from five yards out early on and missed a similar chance before half-time.
Blair Sturrock gave Swindon the lead on the hour when he bundled the ball home after Bees keeper Rob Beckwith failed to hold on to a shot.
Birchall struck in the 85th minute with a right-footed shot from 15 yards straight into the top left-hand corner to set up a replay on Tuesday week.
Swindon: Brezovan, Comminges, Vincent, Ifil, Aljofree, McGovern, Peacock, Easton, Zaaboub (Mohamed 81), Paynter, Sturrock (Roberts 74).
Subs Not Used: Nicholas, Pook, Allen.
Booked: Ifil, Aljofree.
Goals: Sturrock 60.
Barnet: Beckwith, Porter, Gillet, Devera, Yakubu, Puncheon, Bishop, Leary, Carew (Thomas 68), Nicolau, Akurang (Birchall 68).
Subs Not Used: Harrison, Grazioli, O'Cearuill.
Goals: Birchall 85.
Att: 5,944
Ref: Darren Deadman (Cambridgeshire).
BBC Sport Player Rater man of the match: Swindon's Comminges 7.60 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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