Leon Knight scored the only goal of the game but his joy was shortlived after he was later sent off for picking up two yellow cards.
But his 54th minute header was enough to condemn Orient to their first home defeat in League One.
The O's also had defender Justin Miller sent off on 33 minutes for a foul on Darren Pratley and Knight's afternoon of mixed emotions was underlined when Orient keeper Glyn Garner saved his spot-kick.
After Knight's second-half dismissal, Orient poured forward in a bid for a late leveller in stark contrast to the first half when chances were few and far between.
There was little for the crowd to get excited about early on. Orient's top scorer Lee Steele crossed from the left and Gary Alexander headed just wide on three minutes, while at the other end a clever ball into the danger area from Kevin Amankwaah only needed a touch, but three Swansea players all failed to make contact.
The turning point came on 33 minutes when the powerful Pratley broke quickly from midfield as he latched onto Rory Fallon's astute pass to go clear of the Orient back line.
Defender Miller caught the midfielder in the area and was shown red by referee Andy Hall.
From the spot-kick, keeper Garner, who has been called into the Welsh squad, dived to his left to palm away Knight's kick. It was the second successive weekend the O's keeper had saved a penalty.
Orient manager Martin Ling sacrificed Shane Tudor and threw defender Donny Barnard into the fray and, as the game threatened to boil over, both Knight and Orient's Alexander were booked before the break.
In the second half Swansea tried to make their extra man count and Fallon had an effort saved before Knight scored the all-important goal.
The diminutive striker somehow crept in between Orient's central defenders and sent a header into the corner of the net following Alan Tate's throughball.
Knight's joy was shortlived as shortly afterwards he saw red when he picked up his second yellow card of the afternoon.
Orient pressed forward in the closing stages but for all their late pressure all they had to show was two shots from Brian Saah and Steele that were just off target.