Fernandinho volleyed in an 89th-minute winner as City survived Villa scare.
According to Football Association Fair Play delegates, Chelsea the worst-behaved team.
The win closed the gap on fourth-placed Manchester City to four points with six games remaining.
Defeat kept fourth-placed Manchester City nine points behind leaders Chelsea.
Liverpool manager Rodgers said that his club is not looking to sell the 20-year old youngster.
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.