Eddie Howe's side had forced a replay with a 2-2 draw in the first leg, but the Latics triumphed 3-0 to secure a date with West Ham in the fourth round.
The League One leaders took the lead after nine minutes through the Egyptian international who unleashed a powerful shot that breezed past a hapless Artur Boruc.
Substitute Dan Burn doubled the Latics lead after he fired home a left-footed shot from Max Power's assist in the 73rd minute.
Three minutes later, Callum Elder and headed home the third from Power's corner kick to seal the victory for Wigan.
Morsy, who captained his side was making his 24th appearance for Paul Cook's side and has now scored three goals in all competitions.