Juan Mata scored the solitary goal to win a tight Manchester derby as United knocked out EFL Cup holders City to reach the quarter-finals.
The Spaniard pounced to drive low into the corner from Zlatan Ibrahimovic’s pull-back as United set up a home tie with West Ham.
Paul Pogba had hit a post moments earlier as the game came alive in the second period, following a first half in which neither of the bitter rivals managed a shot on target.
City boss Pep Guardiola made nine changes to his side, who are now six games without a win in all competitions – his worst run as a manager.
Their best chance arrived early on when Kelechi Iheanacho headed over from six yards, while second-half substitutes Raheem Sterling and Sergio Aguero made little impact.
Victory was United’s second in five matches, and only Jose Mourinho’s fourth win in 18 meetings with Guardiola.