A stunning game from City, who took control around the 10-minute mark, soon led through Bernardo Silva and never looked back.
The scorer of the opener teed up the second for Mahrez and also had a hand in the third, a Pereira own goal.
United were three down just a quarter of the way into the game.
Only for Rashford to pull one back for the Red Devils in 70th minute. The forward finished clinically into the bottom-left corner.
United must win by two at City in the return fixture to have any hope of progressing though.
The odds are stacked against them, even with that Rashford goal.