Britton pounced from inches out after Cheltenham goalkeeper Shane Higgs had made a diving save from a Lee Trundle left-foot drive.
The goal came towards the end of a highly-entertaining game and made it two wins from the Easter holiday period for the Swans.
With six games to go, the League Two promotion race is still wide open and Swansea are one of five teams in the hunt for the three automatic places including the title.
The match got off to a lively start with a goal after only three minutes and the pace rarely dropped after that.
Swansea took the early lead when a corner taken by Andy Robinson was only partially cleared. Robinson kept possession and headed for the line before delivering a left-foot cross to the far post. Lee Thorpe rose highest to nod his team in front.
Steve Guinan sent a header just wide from a Grant McCann corner for the home side, but Swansea could have increased their advantage on 15 minutes when Trundle crossed from the left and Paul Connor fired over the crossbar from close range.
Cheltenham grew in confidence as the first half progressed and both David Bird, with a thunderous 25 yard shot, and McCann, with a curling free-kick, went close.
An equaliser did arrive after 26 minutes when Cheltenham won possession from a cleared Swansea corner. McCann sent Martin Devaney clear with a pass into space and the winger sprinted the length of the Swansea half before blasting a shot in off goalkeeper Willy Gueret.
It was proving to be a lively encounter in more ways than one with three Swansea players - Kristian O'Leary, Roberto Martinez and Connor - on the end of bookings from referee Steve Tanner.
The second period featured equally committed performances by both sides, although it was Cheltenham who had the better chances to win it.
Guinan had a shot cleared off the line by Kevin Austin and Brian Wilson got clean through with only the goalkeeper to beat but was halted in his tracks by a fine saving tackle from O'Leary.
Higgs pushed a long-range shot from Marcus Bean over the crossbar before Britton swooped to clinch it for the Welshmen.
The afternoon went from bad to worse for the home side, after Grant McCann saw red for kicking out at Lee Trundle.