The Nerazzurri announced the signing via their official website, with manager Antonio Conte landing his top target after it looked as though Lukaku was set to join Serie A rivals Juventus.
Inter tweeted Lukaku in a club shirt as he signalled his arrival:
The Guardian's Fabrizio Romano reported United accepted an offer of £70 million ($85 million) on Wednesday.
The Belgium international has left Old Trafford two years after he joined from Everton for an initial £75 million fee.
Lukaku, 26, scored 42 goals in 96 appearances for the Red Devils. However, he netted just 15 times in 45 appearances across all competitions last term, his lowest tally in a single season since his debut campaign at Chelsea (2011-12) when he failed to score in 12 appearances.
Inter will hope the San Siro can revive the forward, with Lukaku set to team up alongside Argentina striker Lautaro Martinez in attack. Former club captain Mauro Icardi has been frozen out of the squad by new manager Conte and could have played his last game for Inter.
Lukaku could make his debut for Inter when they host Lecce in their opening fixture of the 2019-20 Serie A season on Monday, August 26.