Despite having Scott Canham sent off after only 27 minutes, Leyton Orient never appeared in danger of losing the match.
Swansea only managed two good shots at the Orient goal, with James Thomas going closest with a first half strike that hit Scott Barrett's crossbar.
Canham was booked after 24 minutes and then cruelly after following through Roger Freestone just three minutes later. Freestone made no meal about it, and the red card came as a great surprise.
Referee Paul Armstrong was now committed to book everything and finished with a collection of seven yellow and one red.
While Swansea worked hard, ten-man Orient scored a splendid goal after 53 minutes when Andy Harris played the ball inside a defender and Jabo Ibehre caught on the right.
His cross was inch perfect and striker Chris Tate lost his namesake Alan Tate, to glide the ball wide of Freestone with about the only, real Orient shot of the game.
Orient's Paul Brush had begun his managerial with another 1-0 win at Swansea and commented: "Its nice to get two results here although 1-0 leaves you too nervy.
"I thought we were better with ten men than with 11 as Swansea had dominated the early play.
"We defended well and gradually made them anxious and finally edgy to get a goal back.
"Canham missed controlled the ball and I told him if he had controlled it he would have scored.
"I thought he was unlucky as I don't there was any malice in it." Swansea Director of Football Brian Flynn said: "We did not learn the lesson of how to play against ten men, and hit too many diagonal balls.
"We played them too straight and did not look for room behind them. It makes the game against Bristol Rovers on Thursday a hell of a match. It will be tough and we will be prepared for it."