Lukaku, 26, scored 42 goals in 96 games for United but leaves just two years after his £75m move from Everton.
“Inter were the only club I wanted,” said Lukaku.
“I’m here to bring the Nerazzurri back to the top.”
Lukaku scored 15 goals in 45 games for United last season but was often overlooked in favour of England striker Marcus Rashford by manager Ole Gunnar Solskjaer.
He did not feature in any of United’s pre-season games, with Solskjaer saying the player had an ankle injury.
Belgium manager Roberto Martinez said earlier this summer Lukaku “had to leave” the Premier League club, with Inter boss Antonio Conte indicating the ex-Blues striker was his top transfer priority.
Lukaku’s United career
In his debut season, the Belgian became one of only 11 United players to score 25 goals or more in a campaign since 1992-93. Cantona, Andy Cole and current United manager Ole Gunnar Solskjaer (25 each) are among the strikers who never matched Lukaku’s 27-goal tally in a United campaign.
However, 26% of his United goals – 11 out of 42 – came in the first two months of his Old Trafford career.
Lukaku’s departure leaves United with just three established forwards – Rashford, Anthony Martial and Alexis Sanchez – in addition to youngsters Angel Gomes, Mason Greenwood and Tahith Chong.
United have spent £140m this summer on Harry Maguire, Aaron Wan-Bissaka and Daniel James.