The veteran striker took his tally for the season to 19 with goals in each half as the Seagulls came from behind against mid-table Town.
The visitors made the brighter start and it was no surprise when they grabbed an eighth minute lead. A corner from on-loan Fulham midfielder Michael Timlin was flicked on to the far post where Craig Easton scored with an angled drive.
Brighton, missing first-choice full-backs Matt Richards and Andrew Whing through injury and suspension respectively, had Forster to thank for pulling them level four minutes later.
He picked himself up after he was fouled to meet the resulting free-kick from Dean Cox with a glancing header which gave Petr Brezovan, Swindon's giant keeper, no chance.
Brighton should have been ahead by half-time. Slovakian Brezovan denied them by tipping over Glenn Murray's header and then producing a smart double save to deny Steve Thomson and French newcomer Therry Racon.
The South Coast club struggled to maintain their attacking momentum in the second half but Forster came up trumps once more in the 66th minute.
His curling free-kick from outside the box found the net via Brezovan's right-hand post to stretch Brighton's unbeaten home run in the league to ten matches.
They are now just one place outside the play-offs ahead of trips to Nottingham Forest and Leeds United.
Forster said after the game: "I was delighted to see that free-kick go in. It was a valuable three points - we needed to win.
"It keeps the pressure on the teams above us and we're in a nice position now. We've got two tough away games so let's hope we can pick up some points on the road."