The Toffees will meet Wigan Athletic in the quarter-finals after proving too strong for the League One Latics following the last-gasp 2-2 draw at Boundary Park on February 16.
Belgian striker Kevin Mirallas opened the scoring in the 15th minute with a neat side-footed finish from Darron Gibson’s rightwing cross before Leighton Baines converted from the penalty spot just after the half-hour mark.
Everton were in cruise control when Leon Osman made it three goals to the good in the 62nd minute when he flicked on for Nikica Jelavic only to see the ball go in from his own attempt.
Oldham got one back just over a minute later through Matt Smith, who scored the late leveller in the initial fixture, when he fired a header home from the six-yard box but the damage had already been done.
The Toffees next host Wigan at Goodison on March 9 as they look to secure their first FA Cup since 1995 after losing the 2009 final 2-1 to Chelsea.
2012/13 FA Cup fifth round replay
Goodison Park, Liverpool – 32,688