Stevan Jovetic and Samir Nasri scored either side of halftime for City as Pellegrini finally fell upon a winning formula to get one over his managerial rival having lost both Premier League fixtures this season.
The result completed a difficult week for Mourinho, whose side drew with West Bromwich Albion in the Premier League on Tuesday, and returned City to winning ways after their defeat by Chelsea 12 days ago was followed by a draw with Norwich City.
Manchester City, who had drawn a blank in their previous two games having scored more than a 100 goals this season, will be joined in the draw for the last eight by holders Wigan Athletic and Sunderland.
Ben Watson, who grabbed the winner when Wigan shocked Manchester City in last year's final, continued his love affair with the competition by firing the second-tier side to a 2-1 upset victory over Premier League strugglers Cardiff City.
Sunderland punished Southampton for taking the competition too lightly and making a raft of changes, with a 1-0 defeat thanks to a stunning strike from Craig Gardner.
Sheffield Wednesday's home game with Championship rivals Charlton Athletic was called off because of a waterlogged pitch.
On Sunday, Arsenal host Liverpool, Swansea City travel to Everton and Championship side Nottingham Forest visit third tier Sheffield United.
Championship club Brighton and Hove Albion entertain Hull City on Monday.
(Reporting by Toby Davis; editing by Clare Lovell)
