Chelsea were dumped out of the FA Cup by League One (third tier) Bradford City on Saturday.
Chelsea beat Newcastle United 2-0 for a 10th successive Premier League home win.
Mane gave the hosts a deserved 17th-minute lead, running clear of the defence to slot coolly past Courtois.
Manchester United manager Louis Van Gaal however, was disappointed with his team’s performance.
It was only Sunderland’s third league victory of the season and lifted them to 14th place with 19 points.
Chelsea scored through Eden Hazard and Felipe Luis before Derby pulled one back.
With this win, Manchester United has now won their fifth consecutive match in the English Premier League.
This is Chelsea’s 1st defeat since the second leg of last season’s UCL semi-final against Atletico Madrid.
Rooney has struck five goals in as many starts for Man United against the south-coast club.
Arsenal will also miss the services of Theo Walcott, who is still recovering from a thigh injury.
Chelsea stretched their lead at the top as Sunderland’s splendid resistance at least scattered some hope for nearest pursuers.
Sunderland will have the opportunity to end in-form Chelsea’s run of 19 undefeated matches.
United have struggled in adapting from their usual 4 at the back to Van Gaal’s preferred 3.
The trio has rarely featured for City under manager Manuel Pelligrini and their future with the team looks grim.
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