The Robins were 2-0 up inside 10 minutes thanks to Ben Gladwin's brace, before Michael Smith extended the lead.
Jamie Murphy and Chris Basham also  made it 3-2 and Steven Davies replied to Smith's penalty after the break this how it was.
However, the Swindon held on to win the tie 7-6 on aggregate - setting a new record for the highest number of goals in a play-off semi-final - and Mark Cooper's side will now take on Preston in the League One play-off fina on 24 May.