Erik ten Hag’s side hardly created a chance of note until Bruno Fernandes found Aaron Wan-Bissaka with a superb pass inside the area.
The right-back calmly lifted the bouncing ball to the edge of the six-yard box, where Varane used his height to get above Nelson Semedo and head home.
It was extremely harsh on Wolves, who made light of last week’s change of manager to produce a superb performance in Gary O’Neil’s first game in charge.
O’Neil was booked for his protests in claiming for a late penalty as United clung on for their fortunate victory.
United’s debutant goalkeeper Andre Onana avoided punishment in added time for charging into Sasa Kalajdzic as he clattered into the Wolves striker as he came for a cross without making contact with the ball.
Matheus Cunha should have given Wolves the lead before then as Pablo Sarabia’s low cross found him unmarked just beyond the far post, but despite steadying himself, the Brazilian hit the outside of a post.
Cunha fizzed a low shot wide just before the break, and Pedro Neto fired straight at Onana before Varane’s decisive effort while Fabio Silva had two goal-bound efforts saved.
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