Southampton, beaten 3-2 by United in the League Cup final at Wembley in February, have now won just one of their last eight home games in all competitions while United stretched their winless league run to five.
With United manager Jose Mourinho again resting many of his starters for next Wednesday's Europa League final against Ajax Amsterdam in Stockholm, the visitors were on the back foot for much of the contest.
"We played a good team, a team who tried to win and tried to finish well," Mourinho said after visiting goalkeeper Sergio Romero saved a barrage of difficult shots, including Manolo Gabbiadini's fifth-minute penalty.
"The team wasn't perfect but individually they gave me good things. They showed me they want to be an option for the (Europa League) final."
Romero, second choice to David De Gea in the Premier League but preferred in Europe's second-tier competition throughout the season, seems to have cemented his place for the showdown in Sweden's capital.
With a top-four finish in England out of United's reach, winning the Europa League remains their only route to Champions League football next season and the Argentine keeper shone ahead of the club's most important game under Mourinho.
Having denied Gabbiadini after Eric Bailly handled the ball on the edge of the penalty area, Romero also kept out a fierce Dusan Tadic free kick in the second half before United's Anthony Martial hit the woodwork.
Amid mounting media speculation that De Gea could join Real Madrid next season, Mourinho also expressed hope the Spaniard would stay at Old Trafford when asked when United fans would see him in action again.
"David? I think the first match against (American MLS side) LA Galaxy in preseason," he said.
"I hope to play Joel Pereira against Palace (in the last Premier League game of the season on Sunday) and Sergio in the Europa final."
Having stretched his winless league run to five games as the unimpressive run-in left United certain of finishing sixth, Mourinho can only hope his injury-hit side fire on all cylinders in Stockholm.
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