Anthony Martial and Paul Pogba also struck late on as United outshone all their rivals over the weekend to top the fledgling table, leading manager Jose Mourinho to salute their new-found confidence.
Sunday's other game saw Tottenham Hotspur win 2-0 at newly promoted Newcastle United whose captain Jonjo Shelvey was sent off for a deliberate stamp on Dele Alli in front of the referee.
At Old Trafford, a United side inspired by Nemanja Matic, their 40-million-pound signing from Chelsea, provided another expensive newcomer, the 75-million-pound Lukaku, with the perfect service.
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Lukaku, who also scored against Real Madrid in last week's Super Cup, netted 25 times in the league for Everton last season and looked equally deadly following his move to United.
The Belgian struck in the 33rd and 52nd minutes, first getting on the end of a perfect through ball from Marcus Rashford and then heading a second goal from a drifted Henrikh Mkhitaryan free kick after the break.
Martial and Pogba then ruined former City keeper Joe Hart's return to Manchester with further strikes that sent the crowd away confident that last season's problems of winning at home will not be repeated.
"It was a dream come true," said Lukaku. "The win was good, four goals and a clean sheet.
"Jose Mourinho wants me to be the leader and create as much as I can. The fans want to see attractive football. It is just the start."
Mourinho was not inclined to get carried away, pointing out that United also started last season well, but said: "The team was confident -- a word I like."
At St James' Park, Spurs' victory was almost as easy once Shelvey suffered his moment of madness in the 48th minute.
The spark point was Alli's refusal to give Shelvey the ball after the Tottenham midfielder was fouled. The 25-year-old Shelvey responded with a stamp that left referee Andre Marriner no choice but to give Newcastle's captain the third red card of his career.
Newcastle manager Rafa Benitez said Shelvey apologised to his team mates after costing Newcastle the match on their Premier League return.
Shelvey will now serve a three-match ban.
Goals by Ben Davies and Alli himself sealed the Tottenham win with Kyle Walker-Peters named man of the match on his Premier League debut.
Bizarrely, the full back was a replacement for the almost identically named Kyle Walker, who joined Manchester City in the off-season.
(Editing by Clare Fallon)
