Chelsea went in front when Cesar Azpilicueta pounced on some poor defending to prod the visitors ahead after 25 minutes and Juan Mata doubled their lead with a rasping drive midway through the second half.
Wenger, who enjoyed a simmering feud with the Chelsea boss during the Portuguese's first spell in the Premier League, has not beaten Mourinho in nine attempts.
Manchester United comfortably overcame Norwich City 4-0 with two goals from Mexican Javier Hernandez, and late efforts from Phil Jones and Fabio, but there was an upset for Premier League Fulham who lost 4-3 at second tier Leicester City.
The West Londoners fought back from 3-1 down, but an 89th minute winner from Lloyd Dyer put them out of the competition.
West Ham United beat Burnley 2-0 away in their fourth round match with late penalties from Matt Taylor and Jack Collison.
Second tier Birmingham City came back from 3-1 down to force extra time against Premier League Stoke City.
On Wednesday, Tottenham Hotspur host Hull City in a repeat of their weekend Premier League fixture and Newcastle United welcome Manchester City.
(Reporting by Toby Davis; editing by Ken Ferris)