Charlie Daniels missed a first-half penalty for the Cherries, with Toffees stopper Joel Robles saving the defender's effort which was at a good height for the keeper.
Ross Barkley then netted in the second half with a deflected strike, before Romelu Lukaku struck late on to ensure that the Cherries couldn't make the quarter finals of the competition for just the second time in their history.
Eddie Howe made seven changes from the team that were defeated against Stoke City last weekend with Tommy Elphick starting his first game since the win at Portsmouth in round four.
Both sides started with a real vigour but the first real chance came just before the half hour mark, when Adam Federici was called upon to keep the game goalless. Gareth Barry knocked a ball down the line and Romelu Lukaku showed a devastating turn of pace to beat Elphick out on the left, making his way into the area before prodding an effort from six-yards which Federici had to stop with his leg.
But the Cherries had a glorious chance of their own after 38 minutes when after a good spell of pressure, they were awarded a penalty. Daniels hung up a corner and James McCarthy stuck his arm out to control the ball and Martin Atkinson immediately pointed to the spot.
After his initial corner, Daniels stepped up with a stuttered run up but his firm effort to the right was punched away by Robles.
That was the last notable action of the first-half and the Toffees took the lead ten minutes after the break.
Barkley received the ball on the left-hand side of the area and curled an effort towards the top right corner. The ball took a huge deflection off Dan Gosling and it looped over the back-pedalling Federici into the top corner for the opener.
Howe's men responded excellently to going behind and went so close to the equaliser from a free-kick. Junior Stanislas stepped up from 25-yards and his curling effort took a nick off Lukaku and flew narrowly wide of the right-post with Robles stranded on the other side.
Gosling was the next Cherries man to go close, as he met Stanislas' cross from the left but his header from eight-yards was straight at Robles in the centre of the Everton goal.
The hosts had the momentum and continued to probe the visitors' backline, but the Merseyside club found the crucial second 14 minutes from time after a decent spell of their own.
Bryan Oviedo swung in a low corner from the right which Barry flicked on to find Lukaku unmarked from six-yards to finish with ease.
The Cherries couldn't find a late goal to make it a nervy finish as Howe's men's cup run came to an end.
AFC Bournemouth: Federici, Francis, Elphick, Distin, Daniels, Iturbe (Murray 70), O'Kane, MacDonald (Ritchie 62), Gosling (Smith 76), Stanislas, King.
Unused substitutes: Boruc, Wiggins, Butcher, Pugh.
Everton: Robles, Coleman, Jagielka, Funes Mori, Oviedo, Lennon, McCarthy, Barry, Cleverley, Barkley (Mirallas 80), Lukaku (Niasse 80).
Unused substitutes: Howard, Baines, Stones, Osman, Deulofeu.
Goals: Barkley (55), Lukaku (76)
Attendance: 11,404 (2268 away fans)
Referee: Martin Atinkinson
Yellow cards: McCarthy, Gosling, Iturbe, King