A brace by Andy Robinson (18, 20) set the Swans on their way and when Ferrie Bodde struck immediately after the re-start it was one-way football.
Jason Scotland came off the bench to net in the 77th minute, before Cheltenham gained some consolation when Steve Gillespie's overhead kick reduced the deficit eight minutes from time.
The Swans had Guillem Bauza up front in place of Scotland who was fit enough for a seat on the bench.
Leading the Robins line was former Swan Paul Connor while Alan Wright, who had a loan spell with Cardiff City, was at full-back.
It was Bauza who had the first strike on goal but Shane Higgs went down smartly to save his second minute snap shot.
The home pressure finally told in the 18th minute when a cross from Paul Anderson was headed goalwards by Bauza and Robinson's deflection on the goal-line put the Swans ahead.
Ninety seconds later Robinson caught Higgs unawares when he struck a perfect shot into the far corner from 25 yards after the ball had broken to him in a central position.
A Bodde free kick had the busy Higgs scrambling to tip over as Swansea looked to put the game beyond the visitors before half-time.
It was not until the 40th minute that the Robins won their first corner, but the Swans broke away from Wright's flag kick and Darren Pratley almost scored at the other end.
It only took the Swans two minutes of the second half to increase their lead. Wright missed his tackle and Bodde was left free to fire across Higgs and into the far corner.
In the 73rd minute the floodlights briefly went off, but despite calls from the 267 visiting fans there was enough daylight for referee Paul Armstrong to continue the game.
Scotland went on as a 76th minute replacement for Bauza and within 60 seconds thundered a shot into the net.
Eight minutes from time Gillespie evaded a tackle by Dennis Lawrence and when the ball eventually came into the danger area his overhead kick beat Dorus de Vries in the home goal.