Brentford hit back quickly through Vitaly Janelt's superb volley but an unfortunate own goal by visiting goalkeeper Alvaro Fernandez gave Southampton a halftime lead.
It was only Southampton's third home win of the season and it lifted them above Brentford into 11th place with 24 points.
Brentford slipped to 13th with 23 points.
"It was the first time we've scored four goals at home since November 2017," Hasenhuettl, whose side are unbeaten in four league games, said.
"It's not very often that injury time plays out and we feel comfortable because sometimes it's tight here. Today we had a fantastic performance."
The fixture was supposed to have taken place on Dec. 18 but fell victim to the surge in COVID-19 infections.
Since then the club has been bought by Sport Republic, a London-based investment firm backed by billionaire Solak, and the mood was buoyant on the south coast on Tuesday.
Bednarek just had to get his head to a typically pinpoint delivery by James Ward-Prowse in the fifth minute to give the hosts the launchpad for a vibrant display.
Brentford were not deterred, however, and equalised in stunning fashion as Janelt volleyed in left-footed from a superb cross by Bryan Mbeumo.
Southampton got their noses in front again after 37 minutes in unusual fashion.
As a corner was cleared out to Ibrahima Diallo on the edge of the penalty area, he was given far too much time to deliver a bouncing shot that hit the post but rebounded in off the back of goalkeeper Fernandez's hand.
Albanian forward Broja, on loan from Chelsea, made it 3-1 in the 49th minute when he was played through by Oriol Romeu's through ball and managed to squeeze his shot past Fernandez for his fifth Premier League goal of the season.
Brentford's defence was then caught ball-watching as a long punt forward was well-controlled by Adams who poked past Fernandez with 20 minutes remaining.
It was a disappointing night for Brentford who have impressed during their inaugural Premier League season and the first time they have leaked four goals this season.
"Fair play to Southampton but my job is to look at my team's performance. If we are not playing every second of every Premier League match 100% then we are not getting anything out of it," manager Thomas Frank said.
"We needed to do better at 2-1. We can't give three of the goals away, especially the last two."