On-loan Chelsea striker Broja celebrated his first top-flight start with a decisive second-half finish at St Mary’s to end the hosts’ nine-match winless run.
Saints boss Hasenhuttl had expressed doubt about the 20-year-old’s readiness to begin games at this level having been disappointed with his performance in last month’s Carabao Cup tie at Sheffield United.
But, after earning his country a World Cup qualifying win over Hungary last weekend and with Che Adams injured, Slough-born Albania international Broja justified his selection by capping a scintillating display with a crucial contribution.
“Against Sheffield [United] I was not so happy, today it was much better,” Hasenhuttl said of Broja.
“I think the goal in the internationals definitely gave him confidence and he has something that you can use – he can thread around the box and you also feel when he has the ball he is a threat, he has the speed to go past a player and produce a shot.
“He had two other good scenes where he could score but I must say it was a good performance. I think it’s time for him to show from the beginning that he can play 90 minutes well and today he worked hard.
“Also against the ball it was much better, which in our game is very important. With the ball he showed that he can hurt you and this was good to see.”
Broja powerfully finished a flowing breakaway goal in the 53rd minute by firing into the roof of the net after being teed up by Nathan Redmond.
He hailed the moment as a “dream come true”.
“It was amazing to get my first start in the Premier League, I’ve been dreaming of this moment since I was three years old,” he told Sky Sports.
“Scoring on my first start is a dream come true, I couldn’t ask for anything better. The guys put in a great shift. It was an amazing moment.
“This is the favourite moment of my career so far, hopefully there are many moments and goals more to come.”
Southampton were well worthy of the overdue success, which lifted them a point above their below-par opponents.
Leeds had arrived on the south coast buoyed by a first league win of the campaign after beating Watford a fortnight ago.
But the visitors – deprived of influential duo Kalvin Phillips and Raphinha due to injury and international commitments respectively – rarely threatened.
Leeds head coach Marcelo Bielsa had no complaints about leaving Hampshire empty handed but felt the loss of two more key performers – who joined the likes of Patrick Bamford and Luke Ayling on the sidelines – was significant.
“It was a fair result, we didn’t play well in any moment in the game,” he said. “It is not common to miss six players but the team that started the game was formed of players that usually play.
“We were in condition to play a more even game and thought we could dominate and impose ourselves. That is clearly not what happened.
“To ignore the absentees would not be logical but with the players we counted on, we were in conditions to play a more even game and to take forward our style.”