Moyes has stubbornly refused to discuss his long-term future at Everton in recent months as the Toffees bid to gatecrash the race for Champions League places.
Anichebe started last weekend's 2-2 draw at Tottenham Hotspur and believes that Moyes is helping push the club forward with his inner drive and determination.
He told the Daily Mail: "It's easy for us to get up for every game. That's the way the manager is.
"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."
Everton sit sixth in the Premier League on 52 points, four behind Arsenal in the race for a top-four finish.