City were rocked by Kyle Walker-Peters’ first ever top-flight goal in the opening minutes.
After a poor first half by their high standards, Pep Guardiola’s side kicked into gear after the break.
Spain defender Laporte headed the equaliser, but City could not extend their winning streak in a one-sided finale that saw the visitors lay seige to the Southampton goal.
The champions had to settle for a result that left them 12 points clear of second-placed Liverpool, who have two games in hand and face Crystal Palace today.
For the first time since they were beaten 2-0 by Palace on October 30, City finished a league game without all three points.
“Why should we lose belief in what we have done? It was a good performance. We are going to drop points, but the way we behave is very good,” Guardiola said.
“People will say the Premier League race is not over, but that is good for this side.”
At Old Trafford, Ralf Rangnick’s side were kept at bay until the third minute of stoppage time.
Edinson Cavani crossed for fellow substitute Rashford to score his second goal in successive games.
After a VAR check for offside, the goal was awarded as interim boss Rangnick celebrated consecutive wins for the first time in charge.
It was a crucial victory for United in their fight to qualify for next season’s Champions League via a top-four finish.
United moved one point above West Ham, knocking David Moyes’ men out of the top four.
Tottenham and Arsenal would both go back above United if they win their matches today against Chelsea and Burnley respectively.
“If you score in the last second of the game that is the best type of win, a clean sheet. I think in the end we deserved to win,” Rangnick said.
“Bearing in mind we played a few days ago against Brentford, the team looked physically good.”
Jonjo Shelvey’s second-half free-kick clinched third-bottom Newcastle’s much-needed 1-0 win at Leeds.
Shelvey curled home from 20 yards in the 75th minute, giving Eddie Howe’s side only their second victory of the season.
Newcastle were under pressure for long spells but they held on to climb to within a point of safety with a game in hand on fourth-bottom Norwich.
Leeds are now only seven points above the relegation zone after their damaging defeat.
Everton’s change of management did not result in a change of fortunes as Aston Villa boss Steven Gerrard enjoyed a winning return to Merseyside with a 1-0 victory at Goodison Park.
Emiliano Buendia’s header in first-half stoppage time separated the sides to leave the Toffees just four points above the relegation zone in 16th place.
Duncan Ferguson took temporary charge of Everton following the sacking of Rafael Benitez last weekend after a run of just one win in 13 league games.
Ferguson was unbeaten in three matches as caretaker boss against Chelsea, Manchester United and Arsenal before Carlo Ancelotti took charge just over two years ago.
But he could not replicate that success as former Liverpool captain Gerrard got one over on his old rivals.
Wolves won 2-1 at Brentford in a bizarre game which was delayed after a drone flew over the Community Stadium in west London.
Both teams had to leave the pitch after what was announced as an “unofficial drone” hovered above the stadium, with confused Wolves captain Conor Coady, asking referee Peter Bankes “what’s happening”.
The players eventually re-emerged and the match resumed after a delay of almost 20 minutes.
Joao Moutinho’s curler put Wolves ahead four minutes into the second half before Ivan Toney equalised with a 71st minute volley.
Ruben Neves clinched the points for Wolves in the 75th minute.