Emmanuel Adebayor's 85th-minute header put Arsenal in the last eight of the Carling Cup after Everton played most of the game with 10 men.
Adebayor scored his third goal of the season
Referee Graham Poll, who has been criticised for his handling of Sunday's game between Chelsea and Spurs, sent off James McFadden after 19 minutes.
McFadden was dismissed for dissent after saying something to Poll when he waved away an Everton penalty appeal.
Adebayor met a corner in space to send a second-string Gunners side through.
Everton fans howled "cheat" at Poll as he walked off the pitch.
Poll has now handed out two red cards in the space of four days, after he controversially sent off Chelsea defender John Terry during Spurs 2-1 win over the champions on Sunday.
The referee issued three yellow cards to Arsenal players and ignored a number of penalty appeals from Everton, notably involving Andrew Johnson.
Poll sent off McFadden after Johnson fell in the box under pressure from three defenders.
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After Poll waved away Everton's penalty appeals McFadden clearly said far too much for the liking of Poll, who dismissed the Scottish winger.
Arsenal sent on goalkeeper Mart Poom at the break for Manuel Almunia, the Spaniard still suffering from the kick he received early in the game as Armand Traore cleared under pressure from McFadden.
Poom was quickly into action, saving well from Johnson's hooked volley.
That was a rare Everton chance as the home side piled forward with Mathieu Flamini and the excellent Denilson prominent in the Arsenal midfield.
Arsenal finally got the ball into the net with five minutes to go, Adebayor hurling himself into the box to head home a Flamini corner.
Everton threw on striker Victor Anichebe for Lee Carsley in injury time, but they were unable to send the game into extra time.
Everton manager David Moyes:
"I couldn't have asked for anything more from them and the gut-wrenching thing is to lose to such a soft goal with five minutes to go.
"That was the only disappointing thing - the boys did tremendously well."
Arsenal goalkeeper Mart Poom on his debut for the club:
"I think it was a great performance with a young team and we knew that it would be a hard game.
"Everton are strong at home but I'm very proud to finally make my first appearance for Arsenal.
"And I am happy that I managed to keep a clean sheet."
Everton: Howard, Neville, Yobo, Lescott, Nuno Valente, Osman, Cahill, Carsley (Anichebe 90), Arteta, Johnson, McFadden.
Subs Not Used: Turner, Weir, Davies, Stubbs.
Sent Off: McFadden (19).
Arsenal: Almunia (Poom 46), Eboue, Senderos, Djourou, Flamini, Walcott, Song Billong, Denilson (Randall 70), Adebayor, Traore, Aliadiere.
Subs Not Used: Connolly, Ryan Garry, Perez.
Booked: Song Billong, Aliadiere, Adebayor.
Goals: Adebayor 85.
Ref: G Poll (Hertfordshire).