Marcus Rashford broke Hull City hearts with a stoppage time goal to give Manchester United a late 1-0 win.
Leroy Fer scored with his single shot on target in Swanea City's loss to Leicester City.
Jamie Vardy scored with his only shot on target in Leicester City's 2-1 win over Swansea City.
Joshua King scored a goal with his lone shot on target against Crystal Palace.
Crystal Palace have Scott Dann to thank after he scored a goal in the third minute of stoppage time to secure a 1-1 draw at home to Bournemouth.
Roberto Pereyra didn't start but he did come on at halftime and scored in the second half in his Premier League debut.
Burnley could not keep pace with a lively Chelsea side and hardly threatened to score on the day as they take a 3-0 loss at Stamford Bridge.
Southampton continued its undefeated start to the Premier League season under Claude Puel with a 1-1 draw against Sunderland.
Mesut Ozil created a goal and scored a second as Arsenal sprinted to a 3-0 first half lead and held on to beat Watford 3-1 at Vicarage Road.
Eden Hazard scored the opening goal and it proved to be all Chelsea needed as they outclassed Burnley in a 3-0 romp.
Ashley Williams won a penalty and earned a clean sheet in a big win for Everton over Stoke City on Saturday at Goodison Park.
Sunderland couldn't manage to get its first win of the season but the Black Cats did earn their first point under David Moyes.
Phil Bardsley conceded a penalty to Ashley Williams on a PK and Shay Given was charged with an own goal as Leighton Baines' penalty rebounded in off the goalkeeper for the only goal of the match.
Walter Mazzarri was forced into two defensive changes against Arsenal in game week three with Craig Cathcart and Miguel Britos ruled out for the Golden Boys.
Mark Hughes was forced into a defensive change at Goodison Park on Saturday after Philippe Wollscheid was ruled out because of an inner ear problem.