The match between the two London sides was due to commence at 12:30pm on Saturday afternoon.
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However, with just minutes to go until the scheduled kick-off, thousands of fans could still be seen outside of Selhurst Park.
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One fan on Twitter posted a video on Twitter of the crowds and said: "Gates around the stadium have broken down. 10 minutes before kick off and stadium is only 15% full."
Palace tweeted to say: "We're aware of issues with regard to turnstiles upon entering the stadium, we apologise for any inconvenience caused and are working hard to rectify the situation."
The match eventually got underway 15 minutes later than expected, although some fans were still making their way to their seats.
West Ham took an early lead thanks to Tomas Soucek, who scored his first goal since August.
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However, Palace equalised just minutes later, with Jordan Ayew's low shot finding the back of the net.
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The hosts then took the lead themselves moments later as Wilfried Zaha capitalised on some poor West Ham defending.
Palace made it 3-1 as they approached the half hour mark as Jeffrey Schlupp caught Soucek off guard and slotted through the legs of Lukasz Fabianski.
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The Hammers responded not long after when Michail Antonio headed past Sam Johnstone, with both sides producing an entertaining first half.
An Eberechi Eze penalty put Palace 4-2 up, but West Ham responded moments later with a scrappy close range finish.