Goals from Andy Scott and Daryl McMahon in the first half put the home side in a powerful position and defender Matthew Lockwood added a third shortly after the break from the penalty spot.
Former Orient man Lee Thorpe cut the deficit late on, but the hosts emerged as worthy winners.
The O's who came under fire from manager Martin Ling following their 4-1 capitulation at Shrewsbury, stamped their authority on the game early on despite playing against a first half gale.
Lee Steele and Scott both went close before Orient took the lead before 15 minutes.
Steele was the provider with a clever ball into the path of Scott who smashed home his ninth league goal of the season from 20 yards.
Orient might have had a penalty when Scott went down under a challenge from Leon Britton but it did not take them long to stretch their advantage.
Lockwood sent Steele away down the left, and his left wing cross was swept home by McMahon from close range.
In a frantic end to the first half McMahon was denied by Willy Gueret in the Swansea goal who pulled off a breath taking save.
At the other end Orient keeper Lee Harrison denied both Lee Trundle and Sam Ricketts in quick succession as Swansea stepped up the pace in a bid to get back into the game.
However Orient killed off their comeback hopes early in their second half when goalkeeper Gueret was lucky to stay on after pulling back McMahon as he raced clean through to concede a penalty.
Lockwood rifled home the resultant spot kick and Orient threatened to run riot but City weathered the storm and cut the deficit on 81 minutes when substitute Thorpe, on for Britton fired home his first goal for Swansea following his recent move from Bristol Rovers.