Kristoffer Ajer sealed the win for the hosts in the second half, shuffling past a Saints defender before coolly slotting under Forster's legs.
A clean sheet for Thomas Frank's side meant they recorded their joint biggest victory in the Premier League, equalling the winning margin in their 4-1 win at Chelsea.
Southampton had started the game brightly but faltered after Jansson scored.
They only had one shot on target fewer than the hosts and controlled 60% of the possession, but struggled to convert their chances into goals.
They came closest when Adam Armstrong fired into the back of the net in the first half, but was denied a goal after straying offside.
"We had 19 shots on the goal. The difference was the quality of finishing and defending," Southampton boss Ralph Hasenhuttl told Sky Sports.
"It's nice to play nice football in the box but it's not enough for winning games. It's why we lost today."
His side will also be feeling aggrieved by referee Michael Salisbury's decision not to award Saints a penalty when Armando Broja was seemingly brought down in the box by Brentford keeper David Raya, a decision which was upheld by the VAR.
The visitors' underwhelming performance was met by chants of "you're getting sacked in the morning" and "you don't know what you're doing" which echoed from the away end as Saints fans turned on their manager Hasenhuttl towards the end of the game.
"I didn't hear [them] because I was concentrating on the game," he said of the chants. "I am responsible for everything. When I don't put a winning team on the pitch it's my responsibility."
Saints have now recorded seven defeats in their past 10 Premier League fixtures and have lost nine of their 18 games away from home.
Brentford sit 12th in the Premier League table on 43 points while Southampton are 15th on 40, eight points clear of the relegation zone.