West Ham started the better and almost got the first sight of goal when Cheikou Kouyate got to the byline and crossed for Sakho but it was cut out for an early corner after just two minutes.
West Ham matched the Gunners for the opening half an hour but it was the Gunners who created the first clear cut opening.
Alex Oxlade-Chambelain went on a Maradona-esque run and teed up Carzola at the far post but his cushioned lobbed volley looped onto the roof of Adrian's net.
Bilic's side gave the hosts far too much space on 31 minutes and were almost made to pay as Arsenal went completely unchallenged down the left and the ball fell to Aaron Ramsey who hit a brilliant half volley from 25 yards which took a wicked deflection off Aaron Cresswell and struck Adrian's crossbar.
Arsenal were just getting into their stride and Adrian got lucky when he came to claim a corner but dropped it onto Olivier Giroud's back. He was relieved to see it sit up to smother as he hit the floor though.
Payet's influence on the game was growing for the Hammers and he twisted Arsenal's defence inside out on 35 minutes before unleashing a curling effort which was deflected over for a corner.
Ramsey went close again as the game ebbed and flowed on 39 minutes but should have fed Debuchy on the right who was completely unmarked.
Bilic handed a Premier League debut to 16-year-old Reece Oxford who playing in defensive midfield was excellent keeping Mesut Ozil quiet.
It was the Hammers who took the lead just before half-time when Payet's brilliant free-kick worked its magic.
Kouyate rose above everyone to power home a header leaving Cech, who had come rushing out to punch it clear, with egg on his face on his debut and the visitors 1-0 up at the break.