Second-half goals by Leon Osman and Ayegbeni Yakubu were enough to give Everton an unlikely win over West Brom.
The Baggies were completely on top for the first hour but paid the price for failing to take their chances.
Instead, Everton took the lead when Mikel Arteta played in Osman to fire home and Yakubu headed a second after he beat Scott Carson to a long ball.
Roman Bednar replied from the spot after Phil Neville's handball but West Brom could not find an equaliser.
The result was harsh on the Baggies and, although a first win of the season will please Toffees boss David Moyes, it was easy to see why he is so desperate to add to his squad.
It has been another frustrating week on that front for the Scot, who thought he had agreed a loan deal for Juventus midfielder Tiago, only for the Portugal star to reject the move.
Valencia's giant Serbian striker Nikola Zigic and Espanyol winger Albert Viera are the latest players to be linked with the Toffees but, until new faces arrive, Moyes must make the best of what he has got.
His lack of options at The Hawthorns led to Jose Baxter being given his full debut to become, at the age of 16 years and 198 days, the youngest player to start a competitive match for Everton.
Baxter lined up on the left-hand side of a midfield that also contained the 17-year-old Jack Rodwell, while Neville was forced to play at right-back despite carrying a knock.
So it was perhaps fortunate for Moyes that his side were up against a Baggies side that are still finding their feet at this level.
West Brom enjoyed most of the possession and carved out some decent openings but, on too many occasions, either their final ball went astray or they could not find the finish.
Tony Mowbray's men dominated the early stages but took until the 18th minute to test Tim Howard through Robert Koren's low shot.
Chris Brunt fired just over with a dipping shot from 30 yards but the home side's best chance came and went when Ishmael Miller failed to connect properly with Gianni Zuiverloon's wicked cross and saw his close-range header fly wide.
Everton, with Yakubu looking isolated up front, were struggling to find a response and as the game wore on it seemed only a matter of time until the visitors were broken down.
Yakubu's header was his 100th goal in English football
But that all changed just after the hour mark when Everton took the lead, completely against the run of play.
First, Rodwell thought he had put the visitors ahead when he rose to meet Arteta's corner but referee Rob Styles ruled out his header for climbing.
Seconds later, however, Everton were in front - and Arteta was again involved.
He played an exquisite through-ball to the on-rushing Osman who steadied himself before planting a low shot into the corner of the net.
Baggies tried to respond and James Morrison brought a good save out of Howard after the ball fell to him from a corner.
But Everton struck a killer blow on 75 minutes when Yakubu chased a punt upfield by Joleon Lescott, and beat Carson to the bouncing ball before heading into the empty net.
West Brom continued to come forward but Bednar was their next player to miscue in front of goal after finding himself in space on the edge of the area.
It took a penalty in the 89th minute for them to find the target, Bednar stepping up to finally find the net.
West Brom boss Tony Mowbray:
"Generally we controlled the match for long spells and with lots of possession.
"That is a positive for us, and I cannot remember them having a shot apart from their two goals.
"But we were probably guilty of not creating enough real chances despite seeing a lot of the ball."
Everton manager David Moyes:
"They were much better than us in the first half and maybe they were a bit unlucky not to be in front.
"But the first goal of the game was always going to be the important one and I am glad that we got it.
"We did not do too much else in the game apart from get our goals but we defended well to stay in it."
West Brom: Carson, Zuiverloon, Meite, Barnett, Robinson, Morrison, Kim (MacDonald 71), Koren (Bednar 70), Greening, Brunt, Miller (Beattie 80). Subs Not Used: Kiely, Hoefkens, Cech, Pele.
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