Adrian Colunga scored on his debut as Albion passed a tough Capital One Cup test.
But the Seagulls needed two Jake Forster-Caskey penalties to see off impressive Swindon 4-2 in extra-time.
The sides were locked at 1-1 when former Albion striker George Barker went off with a shoulder injury early in extra-time.
That left the slick-passing hosts down to ten men and the Seagulls seemed to have killed the tie inside the next ten minutes.
Colunga got to the far post to bury a low cross from Adam Chicksen after sub Paddy McCourt had spread the ball to the left.
Then Colunga won a penalty which Forster-Caskey thrashed into Wes Foderingham’s bottom corner.
Swindon had not given up the tie and impressive ex-Seagulls midfielder Yaser Kasim brought them back into with a solo goal on 112 minutes.
They then threw players forward but a mistake by Nathan Thompson let in Forster-Caskey right at the death.
The Swindon skipper brought down Forster-Caskey, who drilled home the spot kick after the red card had been shown.
Albion went ahead after only ten minutes with a brilliant Rohan Ince volley after a move involving Colunga, who was bright throughout.
But Swindon deserved their leveller, given them by Louis Thompson in the opening minute of the second half.