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Ross Barkley and Romelu Lukaku grabbed the goals but Joel Robles was the FA Cup hero for Everton manager Roberto Martinez.
Second-half strikes from Barkley and Lukaku saw the Toffees through to the quarter-finals as 2-0 winners at Bournemouth.
The Cherries were the better side in the first half and had the chance to go ahead seven minutes before the break when James McCarthy handled in the area. But Robles, who kept his place despite regular goalkeeper Tim Howard recovering from a recent injury, dived to his left to make a superb save from Charlie Daniels' spot-kick.
"In the first half, when we were on the ball, I thought was probably our worst performance this season," Martinez said. "We were very sloppy, very disappointing.
"Then the penalty arrives from a corner we didn't deal with. And I thought that's where we saw a very special performance from Joel Robles.
"He was magnificent in the way he controlled the moment, and in the way he enjoys being the goalkeeper of Everton.
"He has been working so hard behind the scenes. Tim has been an incredible influence on him and that showed today."
Everton looked a different side after the interval and made the breakthrough in the 55th minute when Barkley got lucky from the edge of the area.
Dan Gosling attempted to block his shot, only to see the ball loop up off his foot and high into the air before dipping under the crossbar and over keeper Adam Federici.
Lukaku wrapped up the win from close range with 14 minutes remaining after Gareth Barry flicked on Bryan Oviedo's near-post corner.
Martinez, under pressure with his side off the pace in the Premier League, knows this competition is his best chance to salvage something from an otherwise disappointing season.
He therefore took the rare step of naming an unchanged side for the fifth-round trip, whereas Bournemouth, whose priority remains top-flight survival, made seven changes.
But the Spaniard insisted: "There wasn't any need to make changes.
"We have good footballers on the bench and the key of success for us until the end of the season is to see high standards.
"I just feel football will make the decisions, not me rotating and making changes for the sake of it."
Cherries manager Eddie Howe admitted the penalty miss was pivotal, but attached no blame to Daniels.
"He's taken some pressure penalties for us this season," Howe said. "Today he didn't score, but that's the life of a penalty taker I'm afraid.
"There's an element of frustration for us. We were right in game until the first goal.
"Obviously the penalty was a key moment in the game, and it's a cruel deflection for the first goal which has made the difference.
"We wanted to win we wanted and to go through. Ultimately Everton probably had a bit too much quality in the end."