The Red Devils opened the scoring after a blunder from City's new signing Ederson allowed Romelu Lukaku to bag his second goal in as many games for United.
City rallied late to try and salvage a goal against their local rivals, but fall short with their efforts.
Kyle Walker made his debut for City following his £50m move from Tottenham last week.
However, it was City's other new signing that made an impact for Pep Guardiola's side - but not in the way he would of wanted.
A long ball directed to Lukaku drew Ederson out of his goals to try and meet the ball before the Belgian.
The former Benfica goalkeeper got his judgement all wrong as Lukaku nodded past and finished brilliantly from a tight angle to give United the lead.
It resembled a familiar sight for City, who experienced the same defensive frailties last season. Some neat passing tore City to shreds, with Mkhitayran's final ball releasing Marcus Rashford who calmly fired home past Ederson.
The second half saw Guardiola withdraw a number of first-team players, with Mourinho making similar changes after 60 minutes.
The Portuguese will be pleased with his team's intensity and pressure throughout the majority of the game. United come up against Real Madrid in three days time at Levi's Stadium in Santa Clara.
City will also face Los Blancos, but in six days time in Los Angeles.