Manchester City moved to the top of the Premier League table as Tottenham Hotspur and Chelsea secured late wins -- but Liverpool came crashing down to earth at Burnley.
Liverpool won their first match 4-3 at Arsenal, but Burnley ran out 2-0 winners in their first game at Turf Moor after returning to the top flight.
Burnley came out of the traps very strongly and were in front after just two minutes. Nathaniel Clyne gave the ball away and Andre Gray was the provider for Sam Vokes to expertly find the back of the net.
And they doubled their advantage eight minutes before half-time on the counter-attack. Debutant Steven Defour won the ball off Ragnar Klavan and broke forward before feeding Gray, who took one touch before firing home excellently with his left foot.
Leicester City and Arsenal are both without a win in their opening two Premier League matches after they played out a 0-0 draw at the King Power Stadium.
Neither side could be separated on Saturday, leaving them five points off the top of the table.
Arsenal appeared to be the superior team but they were unable to secure the goal needed to lift the pressure that has already built on manager Arsene Wenger.
Leicester had two penalty shouts turned down by referee Mark Clattenburg, first when Danny Drinkwater went down in the box under a challenge by Laurent Koscielny, and then when Hector Bellerin collided with Ahmed Musa.
Manchester City made it six points from two Premier League games under Pep Guardiola as they won 4-1 at Stoke City, with Sergio Aguero and Nolito both scoring twice.
The deadlock was broken after 27 minutes when Stoke defender Ryan Shawcross held Otamendi at a corner and referee Mike Dean pointed to the penalty spot. Aguero strode forward to smash the ball high past Shay Given.
Aguero then got on the end of a fine De Bruyne free kick to head firmly past Given and into the corner.
Stoke were handed a route back into the game just three minutes into the second half when Sterling was adjudged to have obstructed Shawcross, with Bojan thumping the penalty home to halve the deficit.
There was no way back, though, as substitute Nolito put the result beyond doubt -- scoring in the 86th minute and five minutes into stoppage time. That goal was enough to send Man City to the top of the table on goals scored, ahead of Manchester United.
Chelsea looked to be staring defeat in the face as they trailed at Watford with 10 minutes to go, but for the second time in a row Diego Costa both escaped a red card and scored late for a 2-1 victory.
Watford broke the deadlock 10 minutes after half-time. Nordin Amrabat got the better of Cesar Azpilicueta and crossed deep into the box, the ball went over Troy Deeney but fell kindly for Etienne Capoue whose first time shot flew into the net. After failing to score in 36 games for the club, he now has two in two games.
Costa was again lucky to escape being sent off for two yellow cards. Having been sent off in the first half, he appeared to produce a clear dive outside the area but referee Jon Moss decided to take no action.
Moments later Moss angered Chelsea, when he refused to give a penalty for handball by Craig Cathcart.
Chelsea found the equaliser in the 80th minute. Eden Hazard had the initial effort on goal, but Heurelho Gomes couldn't hold on to the ball and substitute Michy Batshuayi popped up to score his first goal for the club.
And the winner came three minutes from full-time as another sub, Cesc Fabregas, threaded Costa through on goal and he made no mistake to send the Blues into fourth.
Victor Wanyama scored a late header to give Tottenham Hotspur their first victory, with another late goal settling matters in a 1-0 win over Crystal Palace at White Hart Lane.
Spurs looked destined for a frustrating afternoon at White Hart Lane as summer signing Vincent Janssen and Harry Kane were both thwarted by a resolute Palace back line.
Instead Wanyama, bought from Southampton in June, capped his home debut with an 83rd-minute winner, heading in from close range to snatch a dramatic 1-0 win.
Hull City's amazing start to the season continued as they made it six points from six with a 2-0 win at Swansea City.
The Tigers were furious that they were not awarded a first-half penalty for a foul on Robert Snodgrass, but they still came away with the victory thanks to two late goals.
The first came in the 79th minute, Snodgrass playing the ball into the area, Curtis Davies nodding it on and Shaun Maloney silencing the home crowd.
And the Tigers made sure of another fantastic win two minutes into injury time as Snodgrass and Maloney linked up well for Abel Hernandez to send Hull into third in the table.
Everton came from behind to win 2-1 at West Bromwich Albion.
West Brom took the lead in the ninth minute. Salomon Rondon forced a good save out of Maarten Stekelenburg, and from the resulting corner the Everton goalkeeper was all at sea and Gareth McAuley headed home at the back post.
The Baggies thought they would be going into the break in front, but Everton levelled in the second minute of added time. Kevin Mirallas and played a neat one two with Ross Barkley before the Belgium international coolly beat Ben Foster.
And Everton turned it around on the hour. Gerard Deulofeu delivered a corner and after Mason Holgate had an effort on goal, Gareth Barry was on hand to head into the roof of the net.