Steve Cooper's men beat Northampton 3-1 at the Liberty Stadium.
Andre Ayew came off the bench to nod in the leveller, before George Byers beat former Swan David Cornell from distance to put the hosts ahead.
And Ayew then sealed the win as he completed a sequence of three Swansea goals in seven minutes.
Four players made first-team debuts - Ben Cabango, Ben Wilmot, Jordan Garrick and Kees de Boer all featured.
Click VIDEO link to watch Steve Cooper's post-match analysis.
Swansea City have been drawn at home to Cambridge United in the second round of the Carabao Cup.
Cambridge knocked out Brentford on penalties in their first round tie.
The game will take place in the week beginning August 26.