Reading should have been home and dry long before Kenny Jackett's side forced them into extra-time with a scruffy late goal.
Kitson's clever lob on 14 minutes had put the Royals in complete control but the home side missed a hatful of chances before Ade Akinfenwa's late strike took the game into extra time.
But Swansea then had Ijah Anderson sent off for a second bookable offence and the ten men were undone by extra-time strikes from the deadly strike partnership of Kitson and Leroy Lita.
Kitson's opener came when debutant goalkeeper Graham Stack launched a long kick forward and Kevin Doyle flicked a header into the path of Kitson, who beat the forward-rushing Willy Gueret with a first-time lob.
Royals captain Steven Sidwell went close to scoring a second five minutes later when neat footwork from Kitson and substitute Glen Little teed him up for a stinging drive that Gueret finger-tipped over his bar.
Little was an early replacement for John Oster, whose full debut lasted only 13 minutes because of injury.
Royals boss Steve Coppell handed three more of his summer signings full debuts as Stack, Steve Hunt and Chris Makin came into a home side seeking a fourth straight victory.
And that had never really looked in doubt against a Swansea side who were themselves without a string of first-choice players because of suspension and injury, including star striker Lee Trundle.
Reading's dominance continued within seconds of the re-start as Gueret was forced to save with his legs from Doyle's clever flick and Kitson was agonisingly close to his second on 53 minutes when he thumped a header against the crossbar from a Hunt free-kick.
Sidwell then saw a flicked header ruled out for offside and Reading's wasted opportunities meant the home side could never be totally comfortable.
So it was not the biggest surprise when Akinfenwa bundled home Anderson's centre. The goal, though, owed much to Paul Connor whose complete miskick in front of goal was enough to distract keeper Stack.
But when Anderson saw red for a rash late tackle Reading were quick to take control again. Kitson coolly slotted in his second of the game after a superb turn and pass from James Harper five minutes into the extra period.
And Harper was again the provider five minutes from time when his cross was clinically turned home by Reading substitute Lita.