The visitors opened the scoring with just two minutes played when Romelu Lukaku curled the ball home from 18 yards.
Here, Sports Mole takes a closer look at the action between the two teams at The Hawthorns.
On target: 1
On target: 4
Was the result fair?
There can be no doubting that Everton deserved to come away from The Hawthorns with the spoils. While they were not always convincing, the Toffees were superior to their opponents in every department. Meanwhile, West Brom were often found labouring and bereft of ideas.
West Brom's performance
Following this poor display, it would be fair to say that West Brom boss Alan Irvine has a big job on his hands if he is to turn the team's fortunes around. The Baggies fell woefully short of the level needed to beat, or even draw with, a side like Everton. There was a lack of organisation in defence, while a paucity of ideas in attack saw them rarely trouble the visitors. Conceding so early in the match did not help their cause, but they failed to make almost anything happen in the remaining 88 minutes. Judging by the reaction of the home fans during and after the match, it would appear that the pressure is already building on Irvine.
The Toffees are sure to be feeling a huge sense of relief after picking up their first win of the campaign. The manner in which they did it was not always pretty, but there were plenty of signs of encouragement throughout against West Brom. What is likely to please manager Roberto Martinez the most is the way that his side controlled the game with their composed passing and moving. The hosts were often made to chase the ball, allowing Everton to dictate the tempo. There were a couple of moments from a defensive point of view that could have been dealt with better, though it was up front where the visitors really thrived. Romelu Lukaku worked well as the front man, with Kevin Mirallas, Steven Naismith and Aiden McGeady providing good support. The start of the season has been a frustrating one for the Merseysiders, but this looked more like the team that often lit up the top flight last term.
Sports Mole's man of the match
Steven Naismith: The Everton midfielder was a model of stamina for his side, chasing back one minute before making a surging run at the other end of the pitch the next. The Scot also provided the visitors with some mettle in the centre of the park, which was needed when the game occasionally turned physical.
West Brom goalkeeper Ben Foster made an absolute howler for Everton's second goal, allowing Mirallas's tame shot to squirm past him and into the net. It was a mistake which effectively cost his side the match because the Baggies were just about hanging in there at that point.
Anthony Taylor made one or two questionable decisions, but his overall display was a solid one. West Brom might not see it that way after being denied a possible penalty in the first half, though the official just about got that one right after John Stones made a risky tackle on Saido Berahino.
West Brom: The Baggies return to action next Sunday when they travel to Tottenham Hotspur in the Premier League.
Everton: The Toffees bgein their Europa League campaign on Thursday as Wolfsburg make the trip to Goodison Park.