Goals from strikers Mikkel Beck and Hamilton Ricard - their first in 1999 - at the end of the first half put the Teessiders firmly in the driving seat, and a third from defender Steve Vickers on 63 minutes sealed the win.
Southampton, still second bottom, missed an opportunity to climb out of the drop zone while Boro halted an alarming slump in form. They had collected just three points from a possible 27 going into the match.
Robson breathed a sigh of relief at his side's first three-point haul of 1999. Suspensions robbed Boro of four key players: Andy Townsend, the captain, Gary Pallister, Paul Gascoigne and Mark Summerbell, all forced to sit the game out.
But the win came at a high price with midfielder Phil Stamp ruled out for a month after damaging a hamstring. Stamp limped off with 20 minutes remaining.
Robson was disappointed to have lost one of his most influential performers on the day.
"You haven't scored first in a game, you haven't won a match, and the longer that goes on, the more the confidence goes out," he said. "But today's results puts that behind us and the lads can look forward to the rest of the season now and play with that bit more confidence.
"I was pleased with the lads today. They knew it was a pressure game, it was a game we couldn't afford to lose. That's even more pleasing when you look back at the performance. A few of my senior pros really stood up to it and enjoyed the challenge today. To me, that's good players who can do it when they are under pressure."
Boro were camped inside the Southampton box on 44 minutes when Gianluca Festa's cross came back off the inside of a post and Beck slotted home.
Little more than a minute later, Paul Jones misjudged Robert Stockdale's long ball and Ricard, without a goal since the 3-2 victory at Manchester United on 19 December, gleefully ended his drought.
Victory was assured with 63 minutes gone when Southampton failed to clear Neil Maddison's corner and Vickers hammered the ball home for his first goal of the season.
"We were lucky that they didn't get two or three more at the end of the day," said Dave Jones, the Southampton manager. "I don't know why we didn't seem up for it, we had a great opportunity to get out of the bottom three today."
Goals: Beck (44) 1-0; Ricard (45) 2-0; Vickers (62) 3-0.
Middlesbrough: Schwarzer; Stockdale, Festa, Vickers, Cooper, Gordon, Stamp (Deane, 71), Mustoe, Maddison, Ricard (Campbell, 83), Beck. Substitutes not used: Kinder, Harrison, Beresford (gk).
Southampton: Jones; Dodd, Lundekvam, Monkou, Colleter, Oakley (Ripley, 60), Marsden, Kachloul, Le Tissier, Hughes, Beattie (Ostenstad, 88). Substitutes not used: Bridge, Hiley, Moss (gk).
Referee: M Reed (Birmingham).
Bookings: Southampton: Le Tissier, Monkou, Marsden, Kachloul.
Man of the match: Ricard.
Attendance: 33,387.Reuse content