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.
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