Tottenham needed an 87th-minute equaliser from Gylfi Sigurdsson to salvage a point against the Toffees on Sunday after an impressive showing from the Merseysiders in the capital. Anichebe revealed afterwards that Moyes' search for perfection was another key reason why his team are now mounting a late assault on the top four.
The Nigerian jokingly admitted the determination of his manager, who is out of contract in the summer, often drives him mad, but he insists the hard work the 49-year-old demands is worth it. "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."
Everton are outsiders to reach the Champions League for the first time in eight years despite their encouraging result at White Hart Lane. Moyes' men are now six points off fourth-placed Tottenham, but crucially the Toffees have a match in hand against Arsenal on April 16 - three days after they face second from bottom QPR.
Everton also have to face Liverpool and Chelsea before the end of the season, but Anichebe insists Champions League qualification is still not out of reach.
"We need to aim as high as we can," said Anichebe, who may well lose his place for the QPR game due to the return of Steven Pienaar and Marouane Fellaini from suspension.
"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.
"We were disappointed we didn't win. It was a really good performance against Spurs. It says a lot about our desire because we're missing two big players.
"Whoever comes in does well - and that just shows what the manager has instilled in us. When you get your chance, you have to take it."