The Toffees sit just four points off the third-placed Gunners after a fine run of four wins from their last five Premier League fixtures. Yet, Everton has also never beaten Chelsea or Liverpool away under Moyes and its last win away to Arsenal came way back in 1996 - but the Scot is hoping to end that run on Tuesday night.
"I've told the players we will have to win one of these big games away from home if we are going to get into Europe," Moyes is quoted as saying by The Guardian. "For the Champions League it may have to be more than one. We are trying to break new ground."
Meanwhile, defender Phil Jagielka accepts the criticism thrown at Everton for its poor away record against the league's top sides - but insists all is not lost if the club fails to win at the Emirates Stadium.
"You couldn't argue against that and say they are wrong because unfortunately we haven't turned up in some really big games," he told reporters.
He added, "That's the challenge now. There is only one way to change that. But if we don't come away from Arsenal with a win we still can't regard it as a negative. We are going to keep kicking on and kicking on."
Jagielka highlighted Everton's FA Cup quarterfinal loss at home to Wigan as a possible turning point in the club's season.
"I watched the Wigan game [while injured] and the boys were not horrific, but for five minutes of the game we were absolutely blown away. Then we couldn't find a goal that would have got us back in it. You have these sorts of games unfortunately," he explained.
"It has happened in a few of our bigger games in the last couple of years and we would like nothing better than to lay that to rest at the Emirates. But it is not always as easy as that."