Wilfried Zaha scored an injury-time equaliser as West Ham this time throw away a two-goal lead.
The Hammers beat Tottenham in the Carabao Cup in midweek after coming back from two goals down at the break.
Back in the Premier League, Javier Hernandez and Andre Ayew netted before the break before Luka Milivojevic pulled one back from the spot and Zaha equalised in the 97th minute for Palace's first draw of the season.
It felt like a win for the Eagles fans.
Mamadou Sakho was out from the Palace defence and his calming influence was missed.
After 31 minute, Ayew fed Aaron Cresswell down the right and his cross was stabbed home with the outside of his right foot by Hernandez for his fourth goal of the season.
In injury-time, Hart made his best stop when he tipped Tomkins' header from a Cabaye corner onto the bar and sub Bakary Sako skied the rebound.
But after sub Michail Antonio had given away possession with seconds to go, Zaah finally beat Hart with a low shot.
Welcome
Greetings folks and welcome to Mirror Football’s live coverage of West Ham’s visit to Selhurst Park to take on Crystal Palace.
The Eagles remain rock bottom of the Premier League, with just three points to their name after nine games and know that a win against fellow strugglers West Ham is vital.
But the 16th-placed Hammers come into the game brimming with confidence following their superb comeback win over Tottenham in the Carabao Cup this week.
The hosts must again make do without Christian Benteke, who will likely miss another month, while their will be late fitness tests for James McArthur and Lee Chung-yong, who have both been on the treatment table this week.
Andy Carroll is available again after his one-match suspension and Andre Ayew will be pushing for a recall following his midweek heroics against Spurs.
Hammers boss Slaven Bilic will be hoping to ride the wave of that win, as his side bounced back from last weekend’s defeat to Brighton in style.
“I know our situation, our situation also determines my situation. All I can say is it is a massively important game and we have to get some points after we lost against Brighton at home,” he said.
“There are rumours... I talk to the chairman, not every day, but often enough and I know my situation and I know our situation. But I am totally focused on the things that I can influence and that is only the training, talking to the players, and the game.
“I am totally focused on the game because only with that in your head can you improve and affect those kind of rumours.”