Sean Dyche had an immediate impact as the new Everton boss, as his side won 1-0 at home to league leaders Arsenal in his first game in charge.
James Tarkowski's 60th minute far-post header following Dwight McNeil's corner was a goal made by two players who had worked with Dyche at Burnley, and the now Everton boss brought some of the work-effort and defensive discipline his former club habitually showed.
The result leaves Arsenal five points clear of Manchester City, who play away to Tottenham on Sunday, while Everton are still in the bottom three after Wolverhampton beat Liverpool 3-0.
Julen Lopetegui's side produced a sensational display to continue their recovery, while there was little sign of Liverpool's awful campaign improving.
Joel Matip's own goal after just five minutes, followed by new signing Craig Dawson's goal in the 12th minute, put Wolves into an early lead and after Liverpool tried but failed to get back into the match, Wolves captain Ruben Neves added a third in the 71st minute.
Manchester United had to dig deep to hold on for a 2-1 win at home over Crystal Palace after Casemiro was sent off with 20 minutes remaining. By then United led 2-0 thanks to Bruno Fernandes' seventh minute penalty and another goal from Marcus Rashford.
Jeffrey Schlupp's snapshot got Palace back into the game in the 72nd minute and there were some nervous moments until Old Trafford was able to celebrate a 13th consecutive home win.
That win lifted United above Newcastle, who were held to a 1-1 draw at home to West Ham. Callum Wilson gave Newcastle a flying start after just three minutes, but Lucas Paqueta saved a point for the visitors just after the half hour.
Brighton move sixth after Kaoru Mitoma's 87th minute header gave them a 1-0 win at home to struggling Bournemouth, while Brentford are seventh after thrashing bottom of the table Southampton 3-0.
Ben Mee and Bryan Mbeumo scored two goals just before halftime, before Mathias Jensen added extra shine to the score with 10 minutes left to play.
Leicester City won a thrilling match 4-2 away to Aston Villa to ease their own relegation worries after twice coming behind after goals from Ollie Watkins and an own goal from Harry Souttar on his Leicester debut.
James Maddison, Kelechi Iheanacho, Tete (another new arrival) and Dennis Praet all scored in an impressive win for the visitors.
Record signing Enzo Fernandez was in the Chelsea side that kicked off the weekend at home to Fulham on Friday night, but he couldn't inspire his side to a win, who were held to a 0-0 draw by an impressive Fulham at Stamford Bridge.