Reports suggested Moyes would face the sack were his side to lose at home to the Toffees but Bowen’s pair of first-half strikes gave West Ham a 2-0 victory.
Despite the speculation over his future, Moyes has appeared remarkably composed in recent weeks, confident both in his team’s ability to turn their form around and of his support from the board.
“That’s the best way to be,” Bowen said. “We know the pressure there is. The manager’s an experienced man, he’s been in the game long enough, he knows what’s at stake and how much pressure footballers are under anyway, so he doesn’t want to add to it.
“But we know, as players, it’s not been good enough. We haven’t been happy with the way we’ve performed.”
The win - West Ham’s first in the Premier League since October - saw the Hammers climb out of the bottom three but Moyes’ job is far from safe, with a tricky run of fixtures against Newcastle, Chelsea and Tottenham to come. Bowen, though, insists it is on the players to make Saturday’s result the start of a resurgence.
"We know as players, we’re not stupid, we’re not daft,” he added. “As players we’ve got ourselves in that position so it’s on us to get ourselves out of it.
"No one’s going to do it for us, no one’s going to roll over and give us easy games. It’s about us sticking together, being strong, resilient and keep plugging away."