Phil Bardsley put the Black Cats ahead in the 12th minute with a fine long-range strike but Everton equalised soon after through Tim Cahill and then went on to dominate a lively encounter.
Royston Drenthe, who had a strong penalty appeal turned down early on, hit the woodwork while Sunderland goalkeeper Simon Mignolet made a fine double save from John Heitinga and Nikica Jelavic late on.
When asked if he was frustrated not to get the victory, Neville told ITV: "I think so, we had home advantage and we wanted to win. It was a great tie to play in.
"We didn't start the game well, but we came into it at the end. I'm disappointed we didn't play more a little bit. Maybe the occasion got to us a little bit early on and we didn't make the most of our chances."
Neville's Sunderland counterpart John O'Shea was pleased to have earned another crack at their opponents at home.
"We're delighted to get them back to the Stadium of Light, we did some great defending, had a bit of luck and a great save from our keeper has kept us in the cup," he said.
O'Shea dismissed any suggestion his side were now favourites though, adding: "Everton are a fantastic team and we are under no illusions just how difficult it's going to be at the Stadium of Light."
Sunderland manager Martin O'Neill felt either side could have won the match.
Asked if he thought his side deserved the draw, he said: "I think we deserved to stay in the cup, it was a fantastic double save at the end (from Mignolet).
"It was a very difficult place for us to come, not only that but we're playing away in the quarter-finals of the FA Cup and from that viewpoint I thought we started off very well, we got a great goal. Of course the equaliser knocked us back a bit and I think after it could have gone either way."
Looking ahead to the replay, O'Neill added: "I think the game is very much in the balance. We have home advantage but that doesn't mean anything, Everton are very very strong."