Rashford stepped up to score with a powerful strike from the spot in the eighth minute. He also hit the bar from a free-kick late on.
After a 4-0 loss at Manchester United on the opening weekend and then a penalty shootout defeat to Liverpool on Wednesday, Chelsea couldn't hold on to the lead given by youngster Mason Mount's seventh-minute goal.
Manchester United has been pursuing the center back for more than a year since his standout performances at the 2018 World Cup.
Vincent Kompany's ferocious long-range shot settled nerves 70 minutes into the nail-biting contest at Etihad Stadium and took Manchester City back to the top of the Premier League.
The defending champions will retain the title if they win their final two games, at home to Leicester and away to Brighton.
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.
Leicester ended a winless run stretching back to January 1 by beating Brighton 2-1 in the Premier League on Tuesday.
Rodgers has been at Celtic since 2016 after a three-year spell leading Liverpool, including a runner-up finish in the Premier League in 2014.
Davison Sanchez, Christian Eriksen and Son Heung-min scored for Tottenham in their 3-1 drubbing of Leicester.
The former England striker slotted a low finish past goalkeeper Jordan Pickford and celebrated his 58th-minute goal with a somersault.
While Vardy's lone goal saw Leicester hand Chelsea their first home defeat this season, Hodgson's Palace stunned Pep's City 3-2 to end their 100 per cent home record.
The game finished 1-1 in regulation time, with Kevin De Bruyne marking the first start of his latest comeback from injury with the opening goal.
Vichai Srivaddhanaprabha, who owned Leicester, was among those killed when his aircraft crashed and burst in flames outside the King Power Stadium.
Leicester's players have been visibly affected by the incident and have spent Monday and Tuesday attending commemorative events to pay tribute to the Thai owner.
Mahrez said it had been "difficult to sleep" since the tragedy that has sparked an outpouring of grief in English soccer and beyond, but added that he always wanted to play in the match.
2 members of Vichai's staff, Nursara Suknamai and Kaveporn Punpare, also died along with pilot Eric Swaffer and passenger Izabela Roza Lechowicz.
The 60-year-old Srivaddhanaprabha, who owns Thai duty-free retail giant King Power and is said by Forbes to be the fifth-richest person in Thailand, bought Leicester in 2010.
It was also their seventh consecutive league victory and moved them up to fourth place, two points behind leaders Manchester City.
Liverpool, the only member of the top six to play on Saturday, moved a point ahead of Manchester United, which hosts fourth-place Chelsea on Sunday.
Giroud, Willian and Pedro Rodriguez, the three-pronged forward line, all scored in the first half against Hull as Chelsea roared into a 4-0 lead by the 42nd minute.
Top News
Latest News
