Cheltenham secured League One survival for the first time in their history last week and the game was played out amid a party atmosphere at Whaddon Road.
Brighton, however, looked set to spoil the celebrations until Robins captain John Finnigan levelled from the penalty spot late in the contest.
Town began the game well enough with Finnigan and lively striker Steven Gillespie both going close with shots from just inside the area.
Gillespie then fired wide after Michael Townsend picked him out with a good through ball, although it was the visitors who took the lead against the run of play in the 23rd minute.
Zesh Rehman delivered a free-kick from the right-hand side and striker Nathan Elder rose highest to send a looping header beyond goalkeeper Scott Brown to register his first ever Football League goal in his first starting appearance.
Alex Revell then went close with a header that would have put Brighton two ahead, but Cheltenham created enough chances in the second half to claim a share of the points.
Brighton goalkeeper Michels Kuipers made a fine save from a Gillespie volley midway through the second half and then fell upon a shot from the same player after he had made an excellent solo run.
The equaliser finally arrived three minutes from time when Brighton's Gary Hart was penalised for handball and Finnigan stepped up to put away the penalty so that both teams could rightfully celebrate with their fans after the final whistle.