Kevin Mirallas, Leighton Baines and Leon Osman all hit the back of the net for the home side in a comfortable victory that earns the Blues a last eight tie with Wigan
Mirallas opened the scoring after 15 minutes with a side-footed half-volley in front of goal before Baines doubled the home side's lead with a successful penalty kick 18 minutes later.
Oldham goalkeeper Dean Bouzanis fumbled Osman's 62nd minute header to gift the home side a 3-0 lead but FA Cup star Matt Smith gave the travelling fans something to smile about with a headed goal of his own almost immediately after the restart.
Everton manager David Moyes made two changes from the side that suffered a 2-1 defeat to Norwich on the weekend, with Steven Naismith and the injured Marouane Fellaini dropping out of the starting XI and Mirallas and skipper Phil Neville coming in.
As for Oldham, caretaker boss Tony Philliskirk named an identical side from Saturday's win over Portsmouth, with former Everton youngster Jose Baxter starting in the centre of midfield.
The underdogs started strongly and attempted to take advantage of Everton's recent vulnerability in defence, but Everton managed to find their feet after a few nervy opening minutes and exerted pressure on the away side with a constant stream of possession.
Oldham looked exposed down the flanks from the start and that proved to be the case when Everton scored their opener after 15 minutes. A perfectly placed Darren Gibson cross found Mirallas in front of goal and the Belgian striker broke away from the static defence and half-volleyed the ball past Bouzanis.
The game's intensity continued in the aftermath of Mirallas' opener when a defensive mix up at the other end allowed Oldham's Baxter to strike the post with a great effort from the edge of the penalty area.
The resulting defensive clean up led to shouts from the away fans for an Oldham penalty after Neville's clearance hit Gibson's arm in the box, but referee Michael Oliver responded to the cries from the Oldham faithful with a shake of the head.
The home side's pressure paid off when another cross caused Oldham problems, this time resulting in a penalty with 33 minutes on the clock after Connor Brown needlessly handled the cross from Seamus Coleman.
Bouzanis got a hand on Baines' poorly placed spot-kick but the limp attempt somehow found its way into the back of the net to give Everton a 2-0 lead to take into the half-time break.
With no signs of improvement on the horizon, Oldham's super-sub and replay winner Matt Smith entered the fray 10 minutes into the second half, and his introduction clearly had a positive effect as the 23-year-old helped the away side carve out their first real chances of the game in quick succession.
But Oldham's pressure was short-lived as Everton made it 3-0 on the 62nd minute after Osman's Steven Pienaar-assisted backwards header was fumbled into the net by Bouzanis.
Not to be upstaged, fan favourite Smith gave Oldham a sense of hope when he headed in from a corner just a minute later to make it 3-1, but the goal proved to be too little too late as Everton held out to book their place in the last eight of the competition.