But West Ham's progress was hampered when they were held to a draw after key midfielder Kouyate was controversially sent off for a challenge on Palace's Dwight Gayle with the Hammers winning 2-1.
The East Londoners had come from behind following a sloppy early goal scored by Palace's Damien Delaney.
And popular former Hammers star Ian Wright has taken issue with goalkeeper Adrian for his part in the goal.
'I can't believe it'
Speaking in his role as a pundit on BBC's Match of the Day, Wright questioned the Spaniard.
"I can't believe what he's done here," Wright told MOTD.
"All you want to do as a goalkeeper here is clear your lines, get it as far away as you can from the goal.
"It's bounced in front of him granted, but he's literally put it on a plate for Delaney to head it in.
"If you look at it, just push that away, both hands, just push it away... you have to say Adrian's at fault there."
Rare but costly error
Adrian has on the whole been superb for the club since signing from Real Betis on a free transfer in 2013, quickly establishing himself as number one.
But he does on rare occasions have the odd erratic or hot headed moment, the sending off against Leicester City in the first home game of the season after rushing up front for a corner being a case in point.
With a very able deputy breathing down his neck in the shape of Darren Randolph, Adrian will need to get back to his brilliant best next time out against Arsenal.