Eddie Howe said Bournemouth's victory over Everton was a "huge moment in our season" as Callum Wilson scored twice at Vitality Stadium
"Psychologically, it was hugely important to win that first home game which is why today was a huge moment in our season," said Bournemouth manager Howe.
Norwich City produced a magnificent all-round display to inflict Manchester City's first Premier League defeat since January at an ecstatic Carrow Road.
Norwich City manager Daniel Farke, speaking to BBC Sport: "Of course it is a special day for us and for the club, against one of the best teams in the world and when we have so many injuries.
Manchester United returned to winning ways and ended Leicester's unbeaten start to the season by edging to a Premier League victory in a pulsating contest at Old Trafford.
After thrashing Chelsea 4-0 on the opening day, United had failed to win any of their last three games but took all the points to inflict Leicester's first defeat of the campaign.
Substitute Jeff Hendrick fired in a stoppage-time equaliser to earn Burnley a point and deny Brighton their first home victory since 2 March.
The Seagulls were set to win it when Neal Maupay volleyed in Solly March's cross at the start of the second half.
Premier League leaders Liverpool shrugged off an early scare to sweep past Newcastle and maintain their 100% start to the season with another dominant attacking display.
Liverpool manager Jurgen Klopp: "I started enjoying it after 35 minutes when we arrived in the game.
Sheffield United boss Chris Wilder says his side "have to wise up quickly" after they were beaten by Southampton in a tight game at Bramall Lane.
The Blades have now won just one of their opening five games on their Premier League return.
Tammy Abraham scored a brilliant hat-trick - and an own goal - as Chelsea destroyed Wolves at Molineux. A day to remember for Abraham, who is joint top of the Premier League goalscoring charts alongside Manchester City's Sergio Aguero
The England forward scored twice in the space of seven first-half minutes after Fikayo Tomori - another 21-year-old - had opened the scoring with a stunning 25-yard finish.
Antoine Griezmann missed his second penalty in four days but France still beat Andorra in Euro 2020 qualifying.
Kingsley Coman rounded goalkeeper Josep Gomes to give Les Bleus the lead, but Gomes then denied Griezmann from the spot.