Mason Greenwood and Brazilian Fred scored the other goals for Ole Gunnar Solskjaer’s side.
Early in the second half Luke Ayling’s superb long-range shot drew Leeds level, but three goals in eight minutes turned the game United’s way in front of over 70,000 fans at Old Trafford.
The 28-year-old Varane, who has played 79 times for France, was described by United as one of “the most decorated players in world football and has amassed over 19 trophies in his career so far, including four Champions Leagues, three Spanish titles and the 2018 World Cup.”
“Manchester United is one of the most iconic clubs in world football and the chance to come here and play in the Premier League is something I couldn’t turn down,” Varane told the United website.
“There is a lot more that I want to achieve in my career and I know I’m joining a squad full of great players who will all have the same determination to win matches and trophies,” added Varane.
“I want to make an impact here and I will give everything to become part of this club’s illustrious history.”
United manager Solskjaer said Varane was a “unique defender.”
“We have great depth of international defenders and he will add his immense skill set and leadership to that group,” added Solskjaer.
Last season, United finished second in the Premier League behind champion Manchester City.