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 supersub Matt Smith claimed his fourth goal of this season's competition, Leon Osman had added Everton's third to all but secured their status as Wigan's quarter-final hosts.
"My feeling is one of pride, first and foremost," Philliskirk said.
"When we went 3-0 down it would have been very easy for people to be thinking it was going to be four or five but I thought we responded magnificently in the second half.
"Ironically, when they scored the third goal we were probably having our best spell in the game.
"It was disappointing to concede that third but the lads kept going, Matt's got another great goal and I thought we played some smashing stuff in the second half.
"You never know, if we'd got another one it would have been an interesting end to the game."