The Nigerian says that the Scottish gaffer is terrible in the lead-up to matches but admits it is all in the best interests of the team who are still gunning for a top-four finish this season.
"It's easy for us to get up for every game. That's the way the manager is," Anichebe said.
"Sometimes he's a nightmare but he's right - every single game is important.
"When I say he's a nightmare I just mean he's so demanding. He's always on you, every single day. That just shows the kind of character he is. He wants to be the best. Hopefully we can fulfil that."
And Anichebe said it was a clear target of Moyes' to guide the Toffees into a top-four position by the end of the campaign.
"We need to aim as high as we can," he added.
"We just need to keep plugging away, win as many games as we can, and we'll be in and around it. We're not far off now. There's definitely a chance."