After a cagey first half, midfielder Hojbjerg gave his side the lead in the 53rd minute, winning the ball high up the pitch and finishing a quick counter-attack by flicking Nathan Redmond's pass home with the outside of his right boot.
The goal prompted a barrage of balls into the box from the home side and Martin Montoya came closest to putting Brighton level, cutting inside and curling a powerful shot that thumped off the crossbar before bouncing to safety.
Brighton pushed Shane Duffy up front for the last few minutes but they failed to get enough quality crosses in to the big Irish defender as Southampton hung on for their third win in their last four games.
Though the performance was punctuated by plenty of errors, Hojbjerg praised the fighting spirit his team have shown.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"Our technical play can be better, (but) performing as a team was excellent. In these away games you don't get a win by being nice - you get it by being brutal and clinical," he told Sky Sports.
"It was the perfect team performance. We're one step closer (to safety), but we're not there yet."
(Reporting by Philip O'Connor, editing by Ed Osmond)
