Sunderland frustrated the Toffees in a full-blooded last-eight clash at Goodison Park, forcing a replay as the game ended 1-1.
Phil Bardsley fired the Black Cats into an early lead but Everton, who levelled through Tim Cahill, had the better chances throughout a lively, although at times scrappy, encounter.
Royston Drenthe had a strong penalty appeal turned down and hit the woodwork, and Sunderland were definitely happy to settle for a draw after their goalkeeper Simon Mignolet produced a brilliant late double save.
Mignolet stretched to keep out a John Heitinga header and then recovered to block from Nikica Jelavic to ensure the teams will meet again at the Stadium of Light.
Neville said: "We knew it was going to be tight. Over the years, every team of Martin O'Neill's we have played we have been neck and neck with.
"We are neck and neck in the league now, there is not much to choose between us.
"You speak to Sunderland, they will feel going back to their place they will be favourites.
"They probably are, but they will be wary of us. The pressure swings to them now.
"Their fans got behind them, it was a great atmosphere, but we are still in the hat and that is the most important thing.
"There weren't too many chances for us to win it - maybe we could have had a penalty, maybe not, then Johnny's header nearly won it.
"But we live to fight another day. It was a great cup tie, let's make the replay a good cup tie as well."
The game started at a frenetic pace, but that dropped in the second half as it became a more stop-start affair.
Bardsley left Everton chasing the game after his long-range strike from a short free-kick hit the bottom corner after 12 minutes, but events might have unfolded differently had Drenthe been awarded an earlier spot-kick.
The Dutchman went down under a challenge from Craig Gardner which referee Andre Marriner did not see as untoward, but replays suggested otherwise.
Neville said: "It was full blooded, it was niggly, there were fouls and shirt-pulling.
"We thought the 50-50s went their way, but it was a really difficult game to referee.
"The good thing was there were no major decisions that were wrong, that is all you can ask for.
"But it is a funny one - I thought Roy was getting fouled early on, but never got any fouls. That probably went against him.
"A couple of their players went down and we didn't get the same protection.
"Then you think, 'come on', but I think a draw was probably a fair result. There was hardly anything in the game.
"It probably would have been an injustice for any team to lose."
Sunderland will have suspended duo Stephane Sessegnon and Lee Cattermole available again for the replay, which has been scheduled initially for March 27.
The Black Cats were also without the injured Kieran Richardson at Goodison and manager O'Neill praised the commitment of his players, particularly defender Bardsley, in coming through a tough test.
O'Neill said: "We are having a few games now and we are having a few problems in the dressing room with injuries.
"Phil Bardsley did magnificently to play, he hurt his ribs in training a couple of days ago.
"Most other people wouldn't have played, but not only did he play, he played superbly for us and scored a great, great goal. But I wouldn't expect much less from him.
"The players dug in deeply, especially in the second half.
"We could have wilted under the pressure but we stuck at it and Mignolet came up with a fantastic double save."