Belgian international Lukaku grabbed his second goal in his second start for United as the Red Devils eased past a disjointed City at Houston's NRG Stadium.
England striker Rashford meanwhile continued his impressive pre-season form with his third goal in three games of United's US tour.
Mourinho hails Lukaku after first United goal
The game was the first-ever Manchester derby played on foreign soil and both teams commemorated the May 22 attacks at a concert in the city shortly before the kick-off.
"It was a very good training session," United manager Jose Mourinho said afterwards. "We had to play well in the first half, is good intensity, the players are tired. I am really pleased and I'm sure Pep [Guardiola] is the same. The result is not the most important thing."
City manager Guardiola meanwhile shrugged off the loss, preferring to enthuse about the performance of teenager Phil Foden, a 17-year-old who has emerged from the club's youth set-up. "I don't have words... It's a long time since I saw a performance like this — it was another level," said Guardiola.
Chelsea, Real agree fee for striker Morata
Earlier, United enjoyed the better of the early exchanges, with Henrikh Mkhitaryan testing new City keeper Ederson with a long-range effort just after 20 minutes.
City should have had a penalty on 31 minutes when youngster Patrick Roberts was tugged back by Chris Smalling as he burst into the area. Instead Smalling managed to get a foot in and clear with Sterling's follow-up effort blocked.
Six minutes later United took the lead with a superb goal from Lukaku. A long clearance from Paul Pogba drew Ederson off his line but Lukaku managed to nod the loose ball past the stranded City goalkeeper before finishing from a tight angle.
Two minutes afterwards, United doubled their lead with another swift counter-attack.
This time it was Mkhitaryan who turned creator, releasing Rashford with a deft pass.
The teenage England striker picked his spot and drilled a low finish across Ederson into the bottom corner.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ