After losing its last league game 9-0 to Leicester, Southampton took a surprise lead in the 13th minute when City goalkeeper Ederson spilled a shot into the path of James Ward-Prowse, who scored on the rebound.
Left-back Andy Robertson charged in from the flank to meet Mane's cross in the 87th minute with Liverpool facing the imminent end of its unbeaten record in the league.
Bournemouth provisionally climbed to sixth in the table with the win, which was sealed by Joshua King's goal shortly before halftime.
Although Liverpool dropped points for the first time since last season in March, Juergen Klopp's side still went six points clear after nine games.
Youri Tielemans scored the winner in the 74th minute to complete the comeback, after Jamie Vardy had canceled out an opening goal for the visitors by former Leicester player Chris Wood.
Alonso broke the deadlock in the 73rd minute at Stamford Bridge as the Blues earned their fifth straight victory in all competitions.
Watford looked headed for its first league win of the season after Abdoulaye Doucoure put the visitors ahead in the sixth minute and Tottenham once again looked short of inspiration as it struggled to find a response.
Bernard scored from a tight angle in the first half and substitute Gylfi Sigurdsson added a late second from distance as Everton beat West Ham 2-0 Saturday in the Premier League to ease the pressure on manager Marco Silva.
United's slim chances of snatching a top-four finish and a place in the Champions League ended in almost symbolic fashion as a defensive breakdown in the 60th minute allowed the hosts to equalize through Isaac Mbenza.
It was Arsenal's first clean sheet in an away league match this season.
The win keeps fifth-placed United two points behind Chelsea, with Tottenham a further point ahead.
At 36 years and 233 days, Phil Jagielka became the oldest scorer in the Premier League this season when he netted from close range in the 10th minute at Goodison Park.
No English team has ever swept the Premier League, FA Cup and League Cup, let alone win European soccer's top prize at the same time.
Ramsey scored in the 30th minute to set Arsenal on its way to a 2-0 win over Newcastle on Monday.
As Ole Gunnar Solskjaer celebrated his first match in charge as permanent manager, the win lifted United to fourth in the standings, one point above the Gunners.
Solskajer can begin planning for the future as he looks to win the Premier League with Manchester United, who haven't been English champions since 2013.
The CEO said that City, currently on top of EPL points table, have been exploring the Indian market for the past two years now.
Manchester City is a point ahead of Liverpool thanks to its 1-0 win at Bournemouth on Saturday and Juergen Klopp's side being held 0-0 by Everton on Sunday.
Ranieri was fired on Thursday (Feb 28) with Fulham in next-to-last place in the league, 10 points from safety with 10 games left.
A second-half penalty from Sergio Aguero secured a 1-0 win for City, which stayed a point back in second place.
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