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.
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.
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.
Ander Herrera also tried his luck from distance shortly afterwards but the young Brazilian gathered comfortably.
At the other end meanwhile, City, who handed a debut to new signing Kyle Walker, slowly got into their stride, with a deflected Raheem Sterling effort tipped over by David De Gea.
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.
United continued to enjoy the better chances early in the second half, with Lukaku almost grabbing a second when his curling strike hit the woodwork on 53 minutes.
Both United manager Jose Mourinho and opposite number Pep Guardiola made a flurry of changes throughout the second half as the game fizzled out.