The Red Devils slumped to a third successive Premier League game without a win on Saturday afternoon at St Mary’s.
Summer singing Daniel James broke the deadlock in the first half with his third goal of the Premier League season.
However, Manchester United were unable to hang on against 10-man Southampton after Saints defender Jannik Vestergaard equalised for the hosts.
The Red Devils have collected five points from their opening four Premier League games to leave Manchester United off the title race pace.
Solskjaer’s side are already seven points adrift of Premier League leaders Liverpool FC following one victory in five outings.
Maguire became the most expensive defender in the world when the Manchester United star completed a £80m move to Old Trafford from Leicester City in the summer.
The record signing believes Manchester United need to rekindle their swagger under Solskjaer.
“Another game where we’ve deserved to win,” Maguire told Sky Sports after the draw.
“We’ve created the majority of the chances and we’ve dropped two points, really.
“The start of our season has been a little bit like that. We deserved to win at Wolves and then we conceded two shots against Palace and they scored two.
“They’ve not really had a chance apart from obviously the one building up to the goal and then the header as well so it’s another set-play, the same as Wolves.
“I feel like we’ve been really fairly solid defensively, but the fine margins are the set-plays and things are punishing us at the moment and we’re not being clinical at the other end.”
Manchester United will take on Maguire’s former side Leicester City in their next Premier League game on Saturday 14 September after the international break.
The Red Devils haven’t won the Premier League title since 2013 when Sir Alex Ferguson was still in charge of the 20-time English champions.
Manchester United finished in sixth spot last term.
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