United were down two-nil at half-time, but were spirited by super-sub Edinson Cavani, who assisted Bruno Fernandes' goal and then scored two himself, with the final one coming in the 92nd minute.
This gave United their eighth consecutive away Premier League win, a run that started at the end of last season.
It also gave them their fourth consecutive Premier League away win this season, all coming after they were trailing early on.
Fernandes also became only the fourth-ever United play to score in five consecutive Premier League away games.
Remarkably, Cavani has already scored 3 Premier League goals, equaling Alexis Sanchez's league tally for Manchester United in just 106 minutes of football.
United now have a very testing match on Wednesday as they face Paris Saint-Germain in the UEFA Champions League, followed by another away game on Saturday, this time against West Ham United.