The Premier League club ended the League One side's memorable run at Goodison Park, largely as a result of a clinical first-half display when recalled forward Kevin Mirallas opened the scoring before Leighton Baines earned breathing space from the penalty spot.
Former Goodison youngster Jose Baxter hit the post for Oldham in between those goals but by the time super-sub Matt Smith claimed his fourth goal of this season's competition, Leon Osman had added Everton's third to secure their status as Wigan's quarter-final hosts.
"It was a tough game," manager David Moyes' No 2 said. "I thought Oldham put a right shift in and pushed us all the way.
"They were a credit to the manager and to themselves. I thought they really acquitted themselves right and made it very tough.
"Having said that, I think we matched them for the attitude and then your quality comes through at the end.
"You saw that with one or two of the goals that we scored and unfortunately we missed a couple of chances to maybe make it a bit more clear-cut."
Darron Gibson's magnificent 15th-minute cross allowed Mirallas to set Everton on their way and Round hailed two of his team's key performers on the night.
"I was very pleased with Kevin," he added. "To come back and show the sort of form he has, it will give him a lot of confidence.
"He has had a real stop-start season with injury and I think we saw signs of him getting back to being sharp and match-ready. His quality finish was what he is about.
"Darron Gibson controls the middle of the pitch for us and I thought his passing was excellent."