The Toffees were able to do that and keep a clean sheet without having their number one goalkeeper, Jordan Pickford, in the side due to injury.
Now that the Merseyside club are set to take on Southampton for their next league outing, Everton manager Frank Lampard has revealed that the shot-stopper is fit enough to feature.
To highlight how impressive the Englishman has been for his side so far this season, Pickford currently has the highest save percentage (83.9%) out of any goalkeeper in the division to have started more than three league games.
Specifically, with 31 shots on target faced, the 28-year-old has saved 24 of them.
In addition to that, the Toffees star has also made more defensive actions outside of his penalty area (11) than any other ‘keeper in the league.
His overall performances have earned Pickford a season rating of 7.26/10 from WhoScored, making him the second-highest rated player at Everton behind Asmir Begovic as a result of his impressive display against West Ham.
With Begovic keeping a clean sheet against the Hammers and making four saves, two punches and one high claim, Lampard now has a decision to make over whether he puts Pickford back in the side knowing he can rely on him, or give the Bosnian another chance to shine.
Bearing in mind how Southampton currently have the third-lowest percentage of shots on target (25.3%) in the league this season, it seems likely that whoever starts in net for the Toffees may not have too much work to do if the outfield players can take control of the game.
Either way, having Pickford available again will surely have lots of Everton supporters pleased and hopeful that he can stay fit for the remainder of the campaign.