Fourth-placed Chelsea suffered a shock 2-0 loss at home to Southampton while Harry Kane and Dele Alli were on target as Tottenham Hotspur came from behind to beat Brighton & Hove Albion 2-1.
Manchester United recovered from a rocky start at home to Newcastle United to win 4-1 with Anthony Martial scoring twice.
Leaders Liverpool are at second-placed Leicester City later.
Italian Ancelotti, a Champions League winner with AC Milan and Real Madrid, was appointed by Everton on Saturday to replace the sacked Marco Silva and was given a warm welcome by the Goodison Park crowd.
A scrappy game was decided 10 minutes from the end when Djibril Sidibe whipped in a cross from the right which Dominic Calvert-Lewin met with a superb angled diving header that flew in off the inside of the far post.
It was a less enjoyable start for former Everton midfielder Arteta, as he took charge of Arsenal for the first time at Bournemouth.
Arteta restored Mesut Ozil to the starting line-up and the German forward looked sharp in the opening exchanges, but chances were rare.
Arsenal’s sloppiness was punished when they gave the ball away cheaply on the left wing and Bournemouth sliced through them quickly, midfielder Dan Gosling stabbing home Jack Stacey’s centre from close range to give the home side the lead.
The Gunners equalised in the 63rd minute as Reiss Nelson’s shot deflected into the path of Pierre-Emerick Aubameyang, and he swivelled to rifle the ball home.
“It was intense, I was so excited, I just wanted to pass that energy to the players and I really enjoyed it. I think we had the chances in the second half to put the game in our favour,” Arteta said.
Chelsea’s poor form continued with goals from Michael Obafemi and Nathan Redmond earned Southampton a surprise victory at Stamford Bridge and condemned Frank Lampard’s side to their fifth defeat in seven league games.
Obafemi broke the deadlock with his first goal of the season, bursting inside from the right and curling into the top corner after a tight opening half hour.
The Saints added their second when Redmond nicked the ball over the advancing Kepa Arrizabalaga to finish a slick attack.
Spurs were also facing a possible upset, until Kane followed up his own shot to equalise in the 53rd minute and cancel out Adam Webster’s first-half header.
Alli got the winner in the 72nd minute with a brilliant lofted finish as he met a Serge Aurier ball.
After losing to Watford last weekend Manchester United needed a response against Newcastle to stay in the frame for a top-four spot but fell behind to Matty Longstaff’s opener.
Old Gunnar Solskjaer’s youngsters hit back in style though with Martial equalising before Mason Greenwood and Marcus Rashford made it 3-1 before the break.
Martial added his second after the interval as United moved into seventh spot, four points behind Chelsea.
In the crucial basement battle substitute Conor Hourihane got the winner for Aston Villa in their 1-0 win over fellow strugglers Norwich City to end a four-match losing run.
Watford moved off bottom spot, above Norwich, with a hard-earned draw at in-form Sheffield United.
Gerard Deulofeu’s goal on the break put Watford ahead but Oliver Norwood’s penalty ensured the points were shared.
A magnificent solo goal in the 90th minute from Jordan Ayew secured a 2-1 win for Crystal Palace at home to West Ham United.
West Ham had grabbed the lead through Robert Snodgrass but former Hammer Cheikhou Kouyate equalised before Ayew’s jinking run and fine finish sealed the points for Roy Hodgson’s side.