City were handed a boost after 90 seconds at the Etihad when West Brom defender Gareth McAuley was sent off.
Costa opened the scoring for Chelsea but Southampton deservedly equalised eight minutes later.
It was Villa’s first win since Dec. 7 and ended a seven-match losing streak in the Premier League.
Rooney scored twice in United’s 2-0 win at the Old Trafford.
Chelsea were held to a 1-1 draw by lowly Burnley as rivals Manchester City thrashed Newcastle United 5-0.
Harry Kane’s brace against Arsenal on Sunday helped Spurs inch closer to Champions League spot.
Arsenal have 42 points from 23 games, two clear of Spurs, and trail fourth-placed Southampton.
Chelsea’s Costa banned for three games for stamping Emre Can; to miss Man City fixture.
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.
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