Non-league Woking battled well at the County Ground but a first half goal was enough to give Swindon a 1-0 win and a ticket to the second round.
Swindon had the majority of possession in the first half but a combination of poor finishing and careless passing restricted the league side to a 1-0 interval lead.
It took 36 minutes for Swindon's superiority to pay off when Jon-Paul McGovern's corner was headed powerfully home by Billy Paynter.
Swindon seemed capable of going up a gear as Paynter went close following a good build up by Simon Ferry, McGovern, Anthony McNamee and Callum Kennedy all involved before Paynter headed just over.
Tope Obadeyi also produced two good saves from Ross Worner in the Woking goal.
At the other end Woking spoilt their few chances by weak finishing. Giuseppe Sole went close with an 18-yard free-kick but it was mostly Swindon.
In the second half it was a more even contest but it followed the pattern of few real threats to either goal.
Despite Woking's desperate attacks in the last few minutes Swindon hung on to take their place in the second round.