Arsenal opened their Premier League campaign with a late-winner against Crystal Palace but Gylfi Sigurdsson struck a 72nd-minute winner for Swansea City as Louis van Gaal's first Premier League game in charge of Manchester United ended in a 2-1 defeat.
Arsenal midfielder Aaron Ramsey pounced as the clock moved past 90 minutes to complete a comeback win for the Gunners who had trailed after 35 minutes to a Brede Hangeland header.
Arsenal equalised on the stroke of halftime when Laurent Koscielny headed in a free kick from new signing Alexis Sanchez.
Earlier, Sigurdsson, back at Swansea after signing from Tottenham Hotspur, fired in from close range to stun the hosts who had drawn level through skipper Wayne Rooney shortly after halftime.
United had plenty of possession in the first half but lacked invention and pace and Swansea took the lead with their first real attack when Ki Sung-yueng slotted home after 28 minutes.
Rooney levelled with an overhead flick after 53 minutes and struck the post with a curling free kick but Swansea took advantage of some poor defending by United left back Ashley Young to pocket three unlikely points.
In the later game, debutant Eric Dier scored in stoppage time to give Tottenham a 1-0 win over West Ham in Mauricio Pochettino first game as manager, with both sides having a player sent off.
Newly promoted Queens Park Rangers lost its first game back in the Premier League as Hull won 1-0 at Loftus Road.
James Chester headed the visitors ahead from close range in the 52nd minute after a cross from Stephen Quinn.
Arsenal 2-1 Crystal Palace
Leicester 2-2 Everton
Manchester United 1-2 Swansea City
QPR 0-1 Hull City
Stoke City 0-1 Aston Villa
West Brom 2-2 Sunderland
West Ham 0-1 Tottenham