The Hammers just need one point from the final game of the season against Southampton on Sunday to secure Europa League football next season, while it would take a nine-goal swing for Everton to deny them a spot in the inaugural Europa Conference League.
The Hammers missed a penalty in the first minute before falling behind at The Hawthorns, but goals from Tomas Soucek, Angelo Ogbonna and Michail Antonio steered them to the brink of Europe.
“I’m delighted we’ve got ourselves in the position, but it is not done yet,” said Moyes. “I’ll go away and sit back and enjoy it, look at the permutations.
“Ultimately, we’re the ones who have to be in charge and win the game. If we win the game then there is no danger.
“At West Ham we’re beginning to find a group who can find ways of winning whatever game it is.
“It was a different type of win tonight, a really tough game for us. They are a much improved side. It was a good victory for us.”