Michael Keane had given Everton an early lead with a well-placed header, but Bowen's finish in the second half ensured West Ham earned a point at Hill Dickinson Stadium.
Moyes admitted afterwards that Everton had missed the chance to put the game beyond their visitors. "We played some decent stuff at times, we got a goal but we couldn't get a second one which would have been all important," he said.
"Once we didn't get it, West Ham grew in confidence and belief that they could get something themselves which they did. Some good things and some not so good things, but West Ham have talented players so it was always going to be a tough game for us."
The Everton manager also acknowledged that facing his former club under new leadership brought its challenges. "I don't know about a different West Ham, I just know the capability of the players," Moyes said.
"We are talking about Brazilian internationals, England internationals, so anybody thinking they haven't got a good team is wrong. They have a very good team and at times they showed it tonight."
Nuno, meanwhile, praised the resilience shown by his new side. "It was a tough game. The reaction was good and overall we competed well," he said. "In the second half we started defending better. Everything starts from there.
"The boys are slowly understanding that a football match is also about how we close our own goal and how many chances we concede. These are things we have to work on."
Reflecting on his debut, Nuno added: "It is about keeping it simple and trying to give the boys a good identity. We will get there in terms of identity and character.
"Everton are a good team in a good moment, so for us this was a positive first step."