The veteran striker struck goals in each half, the second a penalty, to ease the South Coast strugglers out of the relegation zone and above their victims on goal difference.
Brighton boss Micky Adams relegated leading marksman Glenn Murray to the bench after his side had taken only one point from a possible 21. The move paid off as Murray's replacement Stuart Fleetwood had a hand in both goals.
Brighton went ahead two minutes before the break. Matt Richards won possession in midfield before feeding Fleetwood who in turn found Forster to score with a left-foot shot into the far corner.
Brighton doubled their lead seven minutes into the restart. Fleetwood made a nuisance of himself from goalkeeper John Sullivan's long clearance. The Swindon defence failed to deal with the danger and Fleetwood was eventually brought down by skipper Michael Timlin.
Forster made no mistake from the spot to raise his season's tally to ten goals and make sure of a much needed victory for his side.
It was a disappointing start for former Hartlepool chief Wilson. Swindon rarely threatened, although Timlin almost surprised Sullivan with an early free-kick from a wide position which clipped the outside of a post.
Adams said: "The players have had a lot of kicks in the teeth recently but eventually you get a bit of luck yourself. You have to keep believing in what you are doing and work as hard as you possibly can and they are doing that, so I am thrilled for the players more than anybody."
"It was an all round team performance and it won't do our confidence any harm."
Wilson said: "We win together and we lose together. Brighton's game plan was excellent and their two strikers were the best players on the park."