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Moyes succeeded Slaven Bilic in November when West Ham were in the relegation zone and guided the team to safety.
Sam Allardyce was on Wednesday sacked as manager of Premier League side Everton after just six months and despite guiding them away from trouble to an eighth-placed finish, the club announced.
England's all-time record goalscorer with 53 in 119 caps from 2003 through 2016, has netted 10 times for Everton this season.
Jack McLinden, 14, was able to enjoy the once-in-a-lifetime mascot experience through the Norwegian-designed telepresence robot, AV1.
Theo Walcott netted in the second half at Goodison Park to lift Everton into eighth place.
Manchester City now have the opportunity to wrap up the title against Manchester United at Eastlands next Saturday.
Pierre-Emerick Aubameyang's lively debut was aided by three assists from Henrikh Mkhitaryan in his first Arsenal start.
Watford said Everton's approach for Marco Silva had been "the catalyst" for the decision.
Everton signed England winger Theo Walcott from Arsenal in a deal worth a reported 25 million pounds ($34 million, 28.3 million euros) on Wednesday.
Walcott, who has scored 108 goals in 397 matches for Arsenal, is surplus to requirements at the Emirates Stadium having made just 16 appearances this season.
Saturday's late kick-off saw England forward Kane score his 98th Premier League goal and surpass Teddy Sheringham's previous club record of 97.
The 24-year-old England midfielder has signed a five-and-a-half-year contract and will wear the number eight shirt.
Liverpool had led 1-0 at the break after James Milner converted from the penalty spot but Everton looked to have secured a replay when Gylfi Sigurdsson equalised midway through the second half.
Ross Barkley is out of contract at Goodison Park at the end of this season, after rejecting a new deal with Everton.
Jose Mourinho's side were holding onto a slender lead given to them by Anthony Martial when Lingard took possession with nine minutes left at Goodison Park.
Wayne Rooney's hat-trick was instrumental in Everton registering their 2nd win from 13 matches.
Alexis Sanchez's late penalty gave Arsenal a controversial win over Burnley, while managerless Everton endured more misery against Southampton.
Arsenal succumbed to a 62nd-minute penalty from Sehrou Guirassy after the French striker tumbled to ground inside the area under a challenge from Mathieu Debuchy.