Manchester United hold the only 100% record, Tottenham and Arsenal both endured familiar frustrations, while West Ham, Bournemouth and Crystal Palace all lost again.
Here, we look at five things we learned from the weekend's Premier League action.
1. Wenger has work to do
With the future of star man Alexis Sanchez still up in the air, the last thing Arsene Wenger needed was a slow start to the season to unsettle the Chilean further. A couple of late goals helped paper over the cracks against Leicester on the opening day, but back-to-back defeats - including today's 4-0 thrashing at Liverpool - demonstrate just how far the Gunners are from ending their quest for a first title since 2004. Sanchez, who is out of contract next summer, is already unhappy at the lack of Champions League football on offer at the Emirates Stadium and this weekend will have done little to convince him that he should stay in north London.
2. Wood you believe it
Tottenham's Wembley struggles are no secret and they continued on Sunday as Burnley new boy Chris Wood snatched an injury-time equaliser to deny Mauricio Pochettino's men all three points. Spurs won just one of four European games staged at the national stadium last year and they have to play all of their home games there this season. The early signs are not good. Rivals Chelsea left with a 2-1 win last weekend before Wood clinched an unexpected point for the Clarets.
Tottenham are due to host both Real Madrid and Borussia Dortmund in the Champions League in the next few months and they must adapt to their temporary home sooner rather than later.
3. Some dismissals more debatable than others
Referee Mike Dean came in for criticism on Saturday when he sent off Raheem Sterling for a second yellow card following the Manchester City winger's late winner at Bournemouth. While some pundits acknowledged Dean had followed the letter of the law, others were outraged that he opted to punish Sterling for celebrating an understandably joyous moment with the travelling City fans.
By contrast, Miguel Britos could have no complaints following his horror tackle during Watford's goalless draw with Brighton. The Uruguayan defender scythed down Albion's Anthony Knockaert with a dangerous, late lunge and was offered little sympathy.
4. Hammers ready for home comforts
West Ham will be wishing the season could start again. They endured more troubles on their travels on Saturday as they suffered a third successive defeat with a 3-0 reverse at Newcastle. Slaven Bilic's side have been forced to play all three of those games away from their home ground due to the London Stadium being used to stage the World Athletics Championships earlier this month.
Early-season optimism - buoyed by the summer signings of Javier Hernandez, Joe Hart and Marko Arnautovic - is already dwindling among Hammers fans and they will be desperate for their team to return to home turf. After the international break, three of their next four league games will come in front of their own supporters, although the pressure is already building on Bilic.
5. Tough times for De Boer
Crystal Palace boss Frank de Boer admitted he did not expect to lose at home to newly-promoted Huddersfield on the opening weekend and then went on to describe Saturday's clash with Swansea as "must-win". The message clearly did not get through to his players.
Palace, along with West Ham and Bournemouth, remain pointless following the 2-0 loss to the Swans and - perhaps more concerning for Eagles supporters - they are yet to find the back of the net. Dutchman De Boer is quickly discovering what a difficult job he has inherited from Sam Allardyce. He will be determined to bring in some attacking back-up for Christian Benteke before the transfer window closes.