The injury-hit Whites were lacklustre throughout, and failed to muster a single shot on target.
Luke Ayling, Junior Firpo, Kalvin Phillips and Patrick Bamford all missed out through injury, while Raphinha was rested following his recent involvement with the Brazil national team.
An inability to retain possession paired with a regular conceding of fouls made it difficult for Leeds to establish themselves at St. Mary’s.
Early warning signs came in the form of two Saints corners in as many minutes, both of which resulted in Broja beating Diego Llorente to the ball and heading narrowly wide past the far post.
Illan Meslier was forced into action between those two headed chances; a strong diving save denied Nathan Redmond, whose effort from outside the box came after he neatly sidestepped Leeds defenders while cutting inside.
Another corner from the hosts lead to an opportunity for Moussa Djnepo, who was able to get a shot away from the edge of the area after Dallas deflected the initial cross away.
A mistake from Pascal Struijk allowed Broja to engineer a shot from a tight angle inside the area just after the half-hour mark, with Meslier again being called into action at the near post.
Leeds’ first shot of the game, meanwhile, came on the stroke of half-time when Tyler Roberts turned well on the ball but fired well wide from distance.
There was some degree of improvement in the opening stages of the second period, with Jack Harrison’s shot blocked out for a corner.
Moments later though, the Saints were charging down the other end of the pitch, with Redmond ending his run by finding the arriving Broja and leaving him with a simple finish to execute.
Bielsa’s side continued to enjoy some more possession in Southampton’s half, but they lacked the creativity and flair to break through the hosts’ low block.
Dan James used his electric pace to close down Mohammed Salisu, catching the centre-back off-guard and poking the ball just wide of Alex McCarthy’s goal. This proved to be the visitors’ best chance of the game.
This was a day of milestones at Leeds, and not just for Bielsa; Rodrigo made his 50th Premier League appearance, while his replacement Adam Forshaw came on to play in his 50th league match for the Whites.
Unfortunately, neither could turn the ship around, with debutant Joe Gelhardt also struggling to leave his mark on the match.
Wastefulness on Southampton’s part was the only thing that separated Leeds from a thrashing, and Ralph Hasenhuttl’s side thoroughly deserved the three points that enabled them to rise above the Whites in the table.