The England international striker had been tied on 175 with former Arsenal great Thierry Henry.
The win lifts United to fifth place in the standings, two points adrift of Tottenham Hotspur, which occupies the final Champions League spot.
Liverpool, which had the majority of the play on its home pitch, remains in ninth spot with its fifth league defeat under Jurgen Klopp.
But the visitor had David de Gea to thank for fine stops to deny Emre Can and Roberto Firmino, while Jordan Henderson dragged a good chance wide in a drab first half.