The Seagulls, who now go to Shrewsbury, led through a soft penalty from Forster before Livermore settled the match with his first goal for the club.
Brighton went ahead when defender Patrick Kanyuka was adjudged to have brought down Forster in the 24th minute and the veteran striker stepped up to score his first goal for two months.
Kanyuka then needlessly gave away a corner against the run of play and midfielder Livermore headed in Dean Cox’s flag kick at the far post to put Brighton through.
Swindon enjoyed plenty of possession in the second half but struggled to create many clear-cut chances. Brighton’s recalled Dutch keeper Michel Kuipers came to his side’s rescue just after the break by parrying a goal-bound header from Kanyuka following a corner to the far post by Jon-Paul McGovern.
Swindon rarely threatened in the first half and a 20-yard free-kick by McGovern was blocked by the Brighton wall after 32 minutes.