Manchester United s Cristiano Ronaldo takes a shot on goal during the English Premier League soccer match between Manchester United and Southampton at Old Trafford stadium in Manchester, England, Saturday, Feb. 12, 2022. AP
Jadon Sancho's first Premier League goal at home rewarded a bright start from Ralf Rangnick's men.
Yet the Red Devils again failed to build on their lead and were punished when Adams slotted in off the inside of the post two minutes into the second half.
A point edges United level on points with fourth-placed West Ham.
But Arsenal are just a point behind with two games in hand, while Tottenham have played three fewer games and are four points back.
Cristiano Ronaldo was recalled by Rangnick after being dropped to the bench for a disappointing 1-1 draw at bottom-of-the-table Burnley on Tuesday.
That followed a FA Cup exit to Championship side Middlesbrough on penalties after the scores were level at 1-1 after extra time.
Rangnick had to cope with more leaks in the press this week that the United players are unhappy with his coaching techniques.
But the focus of criticism is falling on an underperforming group of star players with the risk of missing out on next season's Champions League growing by the game.
Ronaldo's longest goal drought for 12 years stretched into a sixth game despite having chances he would expect to bury.
The five-time Ballon d'Or winner's best sight of goal came early on when he rounded Fraser Forster, but did not get enough power on his finish to beat Romain Perraud who cleared off the line.
Sancho was by a distance United's best performer of the day and ended his long wait for a Premier League goal at Old Trafford since a £73 million move from Borussia Dortmund.
Bruno Fernandes' pass freed Marcus Rashford down the right and his low cross was turned in at the back post by Sancho.
Southampton did United's top four chances a favour by twice coming from behind to beat Tottenham 3-2 in midweek.
The Saints started slowly after their exertions on Wednesday but gave the hosts a few warnings before the break as David De Gea was forced into saves from Stuart Armstrong and Perraud.
United conceded just two minutes into the second half at Burnley and were caught cold again at the start of the second period when Mohamed Elyounoussi played in Adams, who coolly slotted in off the far post.
Southampton boss Ralph Hasenhuttl worked under Rangnick at RB Leipzig and could easily have got one over on his mentor as twice Armstrong fired wide with good sights of goal before Diogo Dalot cleared Armando Broja's effort off the line.
At the other end Foster also had to be at his best to make a hat-trick of saves in quick succession to deny Harry Maguire, Ronaldo and Dalot.
The Saints stopper saved his best for injury time when he turned away Maguire's header low to his left.
Southampton, though, were well worthy of their point as they made amends for losing 9-0 at Old Trafford last season.
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