Kevin Mirallas settled any early nerves for the Toffees and Leighton Baines fired home from the spot to put the Premier League side in complete control at the interval.
Leon Osman glanced in a third for Everton, meaning a solitary replay from Matt Smith could not prevent David Moyes' men from booking a home date with Wigan Athletic in the last eight.
Despite the absence of injured talisman Marouane Fellaini, Everton began firmly on the front foot. A sharp turn of pace from Mirallas forced Jonathan Grounds to bring him down on the edge of the area, with Baines sending a third-minute free-kick into the wall.
Oldham looked to use the pace of left-winger Jordan Obita in the early stages, but a pair of attempted though balls from Jose Baxter were misplaced when trying to send clear the man who opened the scoring in the first game. That honour on this occasion went to Mirallas, who applied the finish to Darron Gibson's superb cross from the right.
Oldham were almost back on terms immediately as Baxter collected Lee Barnard's lay off to launch an exquisite lofted effort over Everton goalkeeper Tim Howard from 25 yards and against the upright.
James Tarkowski then engaged in an unseemly grapple with Nikica Jelavic as Seamus Coleman delivered from the right by-line but, although Michael Oliver missed that indiscretion, the referee pointed to the spot when the cross struck full-back Connor Brown's outstretched arm.
Bouzanis got a hand to Baines' 34th-minute penalty but there was sufficient power for the ball to slip underneath him and into the net.
The game followed a similar pattern after half-time with Phil Jagielka lashing over when well placed before Osman failed to connect cleanly as Coleman provided further ammunition from the right flank.
In the 55th minute, Latics introduced Smith in place of the ineffective Chris Iwelumo. Their attacks briefly acquired renewed vigour and fellow replacement Robbie Simpson, on for Obita, forced Neville to into a last-ditch clearance behind, while Lee Croft rifled a shot that glanced the side-netting.
But Everton responded by extending their lead slightly fortuitously in the 62nd minute. Steven Pienaar's in-swinging ball from the left was glanced goalwards by Osman, a header that evaded both Jelavic's outstretched boot and Bouzanis' fumbling attempts to gather.
This time Oldham did find a quick response through an increasingly familiar source - Smith allowed far too much space to power home Grounds' corner.
Caretaker boss Tony Philliskirk, on his way to a first defeat in five games at the helm, urged his side forward but that left spaces on the counter-attack, most notably when Gibson should have added late gloss to the scoreline.