Premier League results: Everton and West Ham win at home, Stoke draw with Brighton
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Jese wanted to take it - and Charlie Adam saw his spot-kick saved
Charlie Adam saw his last-minute penalty saved as Stoke missed the opportunity to climb out of the Premier League relegation zone.
The Potters were handed the chance to beat Brighton but Adam's strike was saved by Mat Ryan, before his follow-up was bundled wide.
Brighton had taken the lead when Jose Izquierdo swept home but Xherdan Shaqiri scored for the second game in a row to give Paul Lambert's side a point.
Swansea's good form continued as they clawed themselves further away from danger with a 1-0 win at home to Burnley.
Ki Sung-Yeung's late drive gave Carlos Carvalhal his fourth league win since taking over at the end of December and the Swans moved to 15th in the table.
West Ham and Everton also pulled further away from bottom three with home wins.
Earlier, Harry Kane scored his seventh goal in seven Premier League games against Arsenal to give Tottenham victory in the north London derby.
Swansea's win sees them climb away from danger - for now
The win at Wembley opened up a seven-point gap between the sides, leaving Arsene Wenger's side with plenty to do if they are to secure Champions League football for next season.
Goals were at a premium in the first half of Saturday's afternoon kick-offs, with Izquierdo's superb solo shot for Brighton and West Ham forward Javier Hernandez's header the only strikes.
Marko Arnautovic added a second for the Hammers as they wrapped up the win, while Everton found their shooting boots after the break.
Gylfi Sigurdsson shot home from the edge of the area, Oumar Niasse headed in and Tom Davies bundled in a third before Luka Milivojevic added a penalty for Palace.
Swansea record signing Andre Ayew was reunited on the pitch with his brother Jordan as he made an appearance off the bench in the second half against Burnley, meaning that the Ayew brothers are the fourth set of siblings to play together in the Premier League.
The late drama came at the Bet365 stadium when sub Jese won a penalty for handball. He wanted to take the kick, but Adam was handed the ball and saw his effort saved.
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In the Championship, runaway leaders Wolves moved 13 points clear of the pack after a 2-1 win at home to QPR.
Alfred N'Diaye and Helder Costa had them 2-0 up midway through the first half but they had to hang on after Conor Washington put Rangers back in with a chance.
Will Bristanbul catch on?
At the other end of the table Sunderland produced the comeback of the day.
Their dismal run looked set to continue with a heavy defeat at Bristol City, who were 3-0 up inside 37 minutes. Some Sunderland fans left before half time but Chris Coleman's side fought back to steal a point thanks to a stoppage-time own goal from Marlon Pack.
In Scotland, Celtic progressed into the Scottish Cup quarter-finals as James Forrest struck a hat-trick against Partick Thistle.
Before kick-off there was a minute's silence in tribute to former Celtic midfielder Liam Miller, who died after a battle with cancer aged 36.
Kyle Lafferty scored twice in a decisive display as Hearts beat St Johnstone 3-0, while Highland League side Cove Rangers' cup journey is over after a 3-1 defeat at home to Falkirk.