A return to former club Chelsea had looked likely for much of the summer, only for United to swoop in and gazump the Premier League champions.
Lukaku has arrived in a deal worth £75m (€84.7m) plus add-ons, penning a five-year deal at Old Trafford with the option of a sixth.
“I would like to start by thanking Everton and the fans for the last four wonderful seasons, I have made some special friends and we have shared some amazing moments,” said the striker, who has chosen the number nine worn by Zlatan Ibrahimovic last term.
“However, when Manchester United and Jose Mourinho come knocking at the door it is an opportunity of a lifetime and one that I could not turn down.
“I cannot wait to run out at Old Trafford in front of 75,000 fans but before that pre-season is where the hard work starts and I am looking forward to that first training session.”
That first session came within an hour of official confirmation of the deal, Lukaku having been on holiday in Los Angeles with United midfielder Paul Pogba.
Lukaku stayed with the team at their Beverly Hills hotel on Sunday and arrived with his new team-mates at UCLA the following morning, signing autographs and posing for photos for supporters.
United’s pre-season tour got under way, with the first of a double training session yesterday taking place at Drake Stadium in temperatures around 30 degrees Celsius.
“Romelu is a natural fit for Manchester United,” he said of a player he sold when Chelsea manager in 2014.
United’s first match of the tour comes against the LA Galaxy on Saturday, before friendlies against Real Salt Lake, Manchester City, Real Madrid and Barcelona.
United continue to work on new signings, with Tottenham’s Eric Dier a reported target as talk of a move for Chelsea defensive midfielder Nemanja Matic dissipates.
Inter Milan’s Ivan Perisic continues to be linked with a move to Old Trafford, while speculation remains as to who will follow Wayne Rooney out of the club. United’s all-time top scorer completed his return to boyhood club Everton after 13 years on Sunday.
The omission from the travelling group of Adnan Januzaj, who spent last season on loan at Sunderland, underlines his place in Mourinho’s thoughts. It’s understood Real Sociedad are among those vying for his signature.