Bruno Fernandes struck his second of the game to give the Reds a two-goal cushion following a mistake by Morgan Sanson when the homes side had been threatening to equalise in the first 20 minutes of the second half.
None of the United substitutes warmed up amid Villa’s dominance until just after the hour mark and Rangnick eventually turned to Jadon Sancho at 2-0.
The winger emerged in the stoppage after Jacob Ramsey scored in the 77th minute and he was careless in possession.
Fellow substitute Philippe Coutinho pounced in the 82nd minute to earn Villa a deserved draw.
Rangnick made a double change with the introductions of Donny van de Beek and Jesse Lingard in the 89th minute, but by then United seemed resigned to dropping more points in the Premier League.
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