England captain Harry Kane returned to Tottenham's starting lineup and scored two goals at Goodison Park as Son, Deli Alli and Christian Eriksson crushed the Toffees.
After losing their last two matches following a 22-game unbeaten streak, Unai Emery's side got back to winning ways.
United has been beaten 3-1 by both of its biggest rivals in recent weeks — it also lost by that score at Manchester City last month — and was outplayed on both occasions.
Chelsea moved seven points behind first-placed Manchester City, which it beat last weekend to end the defending champion's unbeaten start to the season.
Charlie Austin's 85th-minute goal sealed a 3-2 win for Southampton in the Premier League and a first win in charge for Hasenhuettl.
Addressing what he describes as "rumours" of Mourinho leaving United, Jorge Mendes says "it's totally untrue. Jose is very happy at the club and the club is very happy with him."
It was the first United-Arsenal game since 1986 that didn't have either United manager Alex Ferguson or ex-Arsenal coach Arsene Wenger in the dugout. Ferguson was in the stands, however.
Arsenal's board may have a big decision to make in a few weeks, with the midseason transfer window opening on January 1.
The team has won one of its 14 league games this season, and that came victory came on September 1.
With Tottenham losing 4-2 at Arsenal later Sunday, Chelsea moved back up to third place.
Chelsea was overtaken by Tottenham in the standings and now is only one point above fifth-place Arsenal, with Manchester City visiting Stamford Bridge early next month.
England's Football Association said the alleged breaches took place in January 2018, when Sturridge was on loan from Liverpool at West Bromwich Albion.
In Shaqiri, Liverpool now has the creative attacking midfielder it has been lacking since selling Philippe Coutinho to Barcelona for 160 million euros (then $192 million) in January.
Completely outplayed by English football's dominant force, Mourinho has little prospect of dethroning City as champions.
For the first time in 130 years of the English top flight, three teams are unbeaten after 12 games. Liverpool remains two points behind City after beating Fulham 2-0 on Sunday and Chelsea is two points further back after being held 0-0 by Everton.
Everton admitted to offering inducements to seven young players, and their families, in a bid to encourage them to register with the club.
De Bruyne, who has only just returned from two months out with ligament damage in his right knee, now has the same problem in his left knee.
In the high octane clash between the two English giants, Barkley robbed United of three vital points after scoring at death, levelling scores at Stamford Bridge.
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.
Alli won the 2015/2016 EPL goal of the season for his shot against Crystal Palace in January 2016.
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