The veteran Czech international moved from Chelsea in a Â£10million summer deal, which was seen as move to provide the Gunners with an edge in the Barclays Premier League title challenge.
However, Cech was caught out when West Ham midfielder Cheikhou Kouyate headed in a first-half free-kick and then Mauro Zarate slotted home a second on the break after 57 minutes to leave Gunners boss Arsene Wenger facing more questions about the direction of his side.
West Ham started with 16-year-old Reece Oxford in midfield, while Olivier Giroud was given the nod in attack ahead of Theo Walcott and Alexis Sanchez made a surprise appearance on the bench following his return from holiday after Chile's Copa America victory.
Arsenal, who won the FA Community Shield at Wembley last weekend with victory over Chelsea, made a bright start with the lively Alex Oxlade-Chamberlain volleying the ball over the crossbar from 20 yards.
The Irons, knocked out of the Europa League qualifying by Romanian outfit Astra Giurgiu in midweek, offered some attacking threat of their own, with Zarate and Diafra Sakho at the top of the diamond fed by the busy Frenchman Dimitri Payet.
Arsenal remained patient with their build-up.
In the 31st minute, Aaron Ramsey's 20-yard half-volley deflected off the back of defender Aaron Creswell and looped onto the crossbar.
The Irons, though, took the lead on 43 minutes.
Payet whipped a deep free-kick into the right side of the Arsenal penalty area, where Kouyate arrived to power a header past Cech, who came flying out to punch but failed to make contact.
The Gunners laboured through the opening stages of the second half, before Giroud fired into the side netting after being played down the left of the penalty area by Mesut Ozil and Iron keeper Adrian then got down quickly to make a fine one-handed save from the French striker.
West Ham doubled their lead on 57 minutes.
Oxlade-Chamberlain failed to clear from the left side of the penalty area, and Zarate picked up the loose ball, turned and slammed a low shot into the bottom corner past a wrong-footed Cech.
Wenger had seen enough, replacing holding midfielder Francis Coquelin with Walcott.
Arsenal continued to press, but lacked a telling pass in the final third while West Ham remained confident when on possession.
Sanchez was sent on with 23 minutes left, and there was a stoppage following a clash of heads between James Tomkins and Giroud in the West Ham six-yard box.
When play resumed, Walcott had an effort blocked before Oxlade-Chamberlain smashed the ball into the side netting.
Arsenal were unable to force their way back into the match during five minutes of stoppage time, as West Ham closed out a first victory over the Gunners since April 2007.
TWEET OF THE MATCH
"What an anti ******* anti climax! So disappointing
We are a million miles away..."
Ian Wright, @IanWright0
Petr Cech: 5
Mathieu Debuchy: 4
Per Mertesacker: 6
Laurent Koscielny: 5
Nacho Monreal: 5
Aaron Ramsey: 7
Francis Coquelin: 6
Santi Cazorla: 6
Alex Oxlade-Chamberlain: 8
Mesut Ozil: 6
Olivier Giroud: 6
Theo Walcott: 5
Alexis Sanchez: 6
James Tomkins: 6
Winston Reid: 6
Angelo Ogbonna: 7
Aaron Cresswell: 6
Mark Noble (c): 6
Cheikhou Kouyate: 7
Reece Oxford: 7
Mauro Zarate: 7
Dimitri Payet: 7
Diafra Sakho: 6
Matt Jarvis: 5
Kevin Nolan: 5
Modibo Maiga: 4
Alex Oxlade-Chamberlain: The Arsenal winger, perhaps thriving as a consequence of his winning goal against Chelsea in the Community Shield, is playing with a previously-unseen confidence
He almost scored a delightful goal, at 0-0, and consistently succeeded when taking on any one of West Ham's defence
He naively contributed to West Ham's second goal but this season could still prove his making capitalising on the absence of Alexis Sanchez to remain in Arsenal's starting XI, particularly with Euro 2016 at the season's end.
MOMENT OF THE MATCH
Petr Cech: Much has been said and written about the strengths of Arsenal's Â£10million summer signing from rivals Chelsea but none could be seen when West Ham took the lead
Dimitri Payet's delivery from a free-kick was exceptional, but the supposedly-commanding Cech was caught out of position when appearing to misread the flight of the ball, meaning Cheikhou Kouyate's rather-routine header ended up as a goal.
VIEW FROM THE BENCH
With so much at stake for West Ham over the coming 12 months, and following their rather embarrassing Europa League exit by Astra Giurgiu on Thursday, animated new manager Slaven Bilic required an impressive result to avoid the type of pressure that can so quickly build, and he got it
There is rightly renewed patience in Arsene Wenger at the Emirates, particularly since retaining the FA Cup in May, but that does not mean he has yet addressed his team's long-term weaknesses.
MOAN OF THE MATCH
Same old Arsenal? Arsene Wenger's team were typically stylish and entertaining but they also wasted multiple goalscoring chances before conceding the most preventable of goals
Thereafter the pressure was on, when an early goal would have likely led to a convincing, one-sided victory
On a separate note, at 2-0 up, West Ham's fans began chanting "are you watching [Sam] Allardyce?"
Did their former manager really do them that much harm?
WHO'S UP NEXT
Crystal Palace v Arsenal (Premier League, Sunday 16 August)
West Ham v Leicester (Premier League, Saturday, 15 August)
Winston Reid hailed defensive midfielder Oxford, who became West Ham's youngest-ever Premier League player and kept a clean sheet on a day to remember.
New Zealand international Reid believes Oxford, 16 years and 236 days old, has a massive top-flight future and told Sky Sports 1: "He's been brilliant.
"He's only 16 but I mean look at him, he's bigger than me and probably better than me!
"But honestly, he's a really good player and he's come into the squad, a young guy, but he listens to us and at 16 he's got his whole future ahead of him."
Oxford admitted it had been "a dream debut", saying: "I had Reidy and Angelo (Ogbonna) behind me just encouraging me, so it was good.
"It's an honour really to play for West Ham and be their youngest player
Hopefully I can just get more games in.
"I found out yesterday night and I was obviously happy."
The Hammers stunned Arsenal as Kouyate's header just before the break and Zarate's second secured a surprise victory.
Reid added: "We had a fantastic game-plan and we followed it.
"We always knew we needed some luck on our side and Adrian made some good saves, but I thought we deserved it, we worked really hard."
Source : PA