A late Stoke City fightback prevented Tottenham Hotspur from claiming a first win of the season in a 2-2 draw at White Hart Lane.
Beaten 1-0 by Manchester United on the opening day, Spurs, who welcomed back goalkeeper and captain Hugo Lloris, went 2-0 up through an Eric Dier header and a deflected Nacer Chadli volley.
But after Spurs striker Harry Kane had been taken off with an apparent injury, Stoke stormed back to level, courtesy of a Marko Arnautovic penalty and a Mame Biram Diouf header.
Meanwhile, Leicester City brought West Ham United crashing to earth and returned to the top of the fledgling Premier League standings with a 2-1 victory at Upton Park on Saturday.
West Ham opened their campaign by winning 2-0 at Arsenal, but they were 2-0 down by half-time as Japan striker Shinji Okazaki headed in his first goal in English football and Riyad Mahrez scored his third goal in two games.
New signing Dimitri Payet replied for West Ham with a powerful strike, but it was not enough to prevent Claudio Ranieri's Foxes reclaiming first place on goal difference from Manchester United.
West Ham's misery was compounded in stoppage time when their Spanish goalkeeper Adrian was sent off for a high foot after going up to contest a corner.
Swansea City also built on an impressive opening performance, following up their battling 2-2 draw at champions Chelsea with a 2-0 win at home to Newcastle United that left Garry Monk's side in fourth place.
Bafetimbi Gomis celebrates with Jonjo Shelvey after scoring the first goal for Swansea City against Newcastle United. (Reuters Photo)
Bafetimbi Gomis rounded Tim Krul to put Swansea ahead in the ninth minute and Newcastle's afternoon took a further turn for the worse shortly before half-time when Daryl Janmaat was sent off for two bookable offences.
After Jefferson Montero and Gylfi Sigurdsson had both made the woodwork shudder, Andre Ayew made the game safe by polishing off a cross from the elusive Montero to register his second goal in two games.
Lukaku at the double
Earlier, Romelu Lukaku scored two first-half goals and Ross Barkley added another as Everton claimed their first victory of the season by winning 3-0 at Southampton.
Belgium striker Lukaku headed home Arouna Kone's 22nd-minute cross and swept in from Barkley's astute pass on the stroke of half-time before the England midfielder neatly curled in a third goal six minutes from time.
Everton's Belgian striker Romelu Lukaku (L) jumps to score the opening goal of the English Premier League football match between Southampton and Everton at St Mary's Stadium. (AFP Photo)
Everton's first victory at St Mary's since 2002, elevated Roberto Martinez's team to third place in the table, two points below Leicester and Manchester United, who won 1-0 at Aston Villa on Friday.
"We've found it tough to come here in recent years, but I thought the performance today was outstanding in every department," said Martinez. "We scored three very good goals and could have had more."
Sunderland's early-season woes continued as they went down 3-1 at home to promoted Norwich City, leaving Dick Advocaat's side at the foot of the table.
Goals from Russell Martin, Steven Whittaker and Nathan Redmond put Alex Neil's visitors in command at a sombre Stadium of Light, before 21-year-old substitute Duncan Watmore replied for Sunderland.
Watford, another promoted side, remain unbeaten but also winless after a goalless draw at home to West Bromwich Albion.
Manchester City host Chelsea in an early clash of title contenders on Sunday, when the visitors will hope to move on from the furore sparked by manager Jose Mourinho's demotion of medics Eva Carneiro and Jon Fearn.