The Seagulls, without a win in nine league matches before the start, looked low on confidence, but boss Micky Adams will have been pleased with the way his players stuck to their task.
Cheltenham, whose positive mid-season run now appears to have come to a grinding halt, were the better side for much of the contest but never convinced anyone they could go on and win it.
The Robins made a dream start when former Brighton midfielder Ian Westlake drove a shot from the edge of the area through a crowd of players with goalkeeper John Sullivan unsighted.
Cheltenham were unable to press home their advantage despite some untidy defending from the visitors and when a second goal did come it had an element of fortune about it.
Alex Russell's corner from the left appeared to hit a Brighton defender in the box, but Chris Westwood was also on hand to get a touch and claim his second goal in two matches since arriving on loan from Peterborough United.
At that point Town should have been home and dry, but there was a nervousness about them and when Martin Allen replaced striker Elvis Hammond with defender Andy Gallinagh 15 minutes from time, it seemed he had settled for a rearguard action in the closing stages.
Veteran striker Nicky Forster gave the Seagulls hope with a looping header from a Tommy Fraser cross in the 78th minute; and although Cheltenham's Ashley Vincent placed a good chance wide shortly afterwards, the momentum remained with Brighton.
Two minutes into time added on a corner, awarded by referee Chris Sarginson amid protests that the ball had come off a Brighton player, was taken by substitute Dean Cox.
Defender Adam Hinshelwood stole in at the near post to power a header past goalkeeper Scott Brown and carve out a valuable point for his team.