Tottenham Hotspur were denied a first win of the season as Stoke City battled back from two goals down to secure a 2-2 draw.
Spurs, who lost 1-0 at Manchester United in their opening game, took a lead on 19 minutes when Eric Dier headed home a Christian Eriksen corner.
Stoke should have been level when Mame Biram Diouf headed straight at Spurs keeper Hugo Lloris on 44 minutes, and they were punished shortly afterwards when Nacer Chadli volleyed home for 2-0.
The Potters were given hope, though, when Marko Arnautovic converted a penalty in the 77th minute after Toby Alderweireld fouled Joselu.
Six minutes later, Stoke were level when Diouf went to head home Stephen Ireland's cross and the ball found its way past Lloris.
Leicester City went top of the Premier League with a 2-1 win over West Ham United.
Shinji Okazaki, who arrived from Mainz this summer, opened his Leicester account when he headed home after West Ham keeper Adrian had saved his volley.
Marc Albrighton then played a superb ball across the box for Riyad Mahrez, who scored a brace in last week's win over Sunderland, to double their advantage on 38 minutes.
However, the Hammers fought back and, having seen a strong penalty shout turned down when Diafra Sakho appeared to be fouled in the area, they pulled one back on 55 minutes through Dimitri Payet's fierce strike.
The Hammers then saw Adrian sent off in injury time for a high boot, forcing defender Carl Jenkinson to go in goal for the final moments.
Sunderland's disastrous start to the season continued as they lost 3-1 at home to Norwich City.
Having lost 4-2 to Leicester on the opening day, Dick Advocaat's men had looked to bounce back against newly promoted Norwich but found themselves behind on 26 minutes when Russell Martin deflected the ball home after a save from Costel Pantilimon.
The home fans started making their way for the exit on 37 minutes when Steven Whittaker made it 2-0 after a one-two with Wes Hoolahan.
Nathan Redmond then deepened their misery took advantage of some poor defending to add a third on 57 minutes.
Duncan Watmore scored a late consolation for Sunderland on his Premier League debut.
Newcastle United are still searching for their first win under Steve McClaren as they lost 2-0 at Swansea City.
The Swans were ahead on nine minutes when Bafetimbi Gomis finished after a fine assist from Jonjo Shelvey.
Newcastle -- held to a 2-2 draw by Southampton in McClaren's first game in charge -- then saw their prospects worsen shortly before the break when Daryl Janmaat picked up a second yellow card.
Summer signing Andre Ayew then made it two goals in two games when he headed home a Jefferson Montero cross.
In the day's early kickoff, Romelu Lukaku scored a brace as Everton beat Southampton 3-0 at St Mary's.
Everton had drawn their Premier League opener 2-2 against newly promoted Watford, and a banner was flown above the stadium before kickoff protesting against chairman Bill Kenwright.
Banner flown over St Mary's, where it remains goalless between #SaintsFC and #EFC. pic.twitter.com/ne0c9hDIhi
- ESPN FC (@ESPNFC) August 15, 2015
However, Lukaku gave Everton the lead after 22 minutes when a fine counterattack saw him bullet home a header from Arouna Kone's outstanding cross, and he doubled their lead seconds before half-time, shooting low into the corner.
Graziano Pelle and Sadio Mane had chances for Saints in the second half, but the visitors rarely looked troubled, and Barkley wrapped up the scoring on 84 minutes when he curled past goalkeeper Maarten Stekelenburg for his second goal in two games.
West Bromwich Albion registered their first point of the season after frustrating Watford in a goalless draw at Vicarage Road.
Tony Pulis' side very nearly nicked all three points towards the end but Saido Berahino, still being linked with a move away before the end of the month, fluffed his lines in front of goal.
A West Brom would have been harsh on Watford, though, as they were for long periods the only team interested in attacking.