Jose Mourinho's side also lost striker Romelu Lukaku to a serious looking head injury as they were forced to settle for a point for a third game in succession.
The hosts had a strong penalty appeal turned down and had a late Paul Pogba goal disallowed for offside, but Saints arguably deserved a point for their resilience.
Before he went off after a nasty clash of heads with Wesley Hoedt, Lukaku headed over a glorious early chance for United, while Jesse Lingard also put a header narrowly wide.
Lingard also appeared to be frustrated by the hand of Maya Yoshida in the box, but nothing was given by referee Craig Pawson.
The Lukaku injury came less than 10 minutes into the game. The striker was carried off on a stretcher and wearing an oxygen mask after a stoppage of around five minutes.
That further changed Mourinho's game plan after leaving Zlatan Ibrahimovic out of his squad altogether. Marcus Rashford, dropped to the bench, was soon sent on.
Saints boss Mauricio Pochettino also made a big call by dropping goalkeeper Fraser Forster.
Forster's replacement Alex McCarthy looked confident as he made his first Premier League appearance for almost two years, but he was fortunate not to concede early on to Lukaku. The £75million striker headed over from seven yards after being picked out by Juan Mata.
It proved Lukaku's only significant involvement, but Southampton went close soon after as Sofiane Boufal beat the offside trap and pulled the ball back through a crowded area to James Ward-Prowse.
Ward-Prowse made a good connection, but David de Gea saved well to his left.
The atmosphere fell flat as the first-half wore on with neither side able to fully impose themselves.
United had the better of the openings, but were wasteful in possession and Henrikh Mkhitaryan, back in the starting line-up for the first time since November 5, was particularly culpable.
On the occasion Mkhitaryan did deliver a good cross, Lingard headed narrowly wide.
Saints also survived a penalty appeal for handball against Pierre-Emile Hojbjerg, as well as the one against Yoshida, while McCarthy saved from Mata and Nemanja Matic.
At the other end Ward-Prowse forced De Gea to save from a free-kick and an unmarked Hoedt headed wide following a corner.
Shane Long looked set to end a goal drought stretching back to February early in the second half, but De Gea saved with his feet.
Mourinho began to get frustrated and shouted angrily at fourth official Bobby Madley after apparently disagreeing with a decision.
His team struggled to match that fire, but Ashley Young went close with a free-kick deflected narrowly wide.
United applied near constant pressure in the closing stages, but Saints managed to hold firm.
Pogba did tap the ball into the net from close range after a goalmouth scramble, but the flag was raised for offside.
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