Swans top scorer Lee Trundle ended a personal barren run by getting his first league goal since the end of December in the 12th minute, but Wayne Purser hit back for Leyton Orient in the 31st. City weren't to be denied three points though as defender Alan Tate got the winner with 9 minutes to go.
At Plainmoor, Torquay followed Saturday's win at Hull City with victory over the Tigers' Humberside rivals Scunthorpe. Matt Hockley's 13th minute effort was the only goal of the game to make it 6 wins from 7 for the Gulls, who are up to 6th.
The other match in Division Three this evening ended in a 1-1 draw between Bury and Southend. Leon Constantaine opened for the visitors in the 27th minute, but Gareth Seddon's last minute goal saved a point for the Shakers.