Mignolet saved from John Heitinga and Nikica Jelavic in quick succession as time ran out in a full-blooded quarter-final clash at Goodison Park that finished 1-1.
Sunderland had led early after a superb strike from Phil Bardsley but Everton replied through Tim Cahill and had the better of the chances throughout with Royston Drenthe denied a penalty and hitting the woodwork.
O'Neill said: "I think we started the game brilliantly, got the goal and we were playing very well.
"The equaliser knocked us back, we lost a bit of confidence with it. There was a big surge from the home crowd.
"I thought the second half could have gone either way but they pressed us more towards the end of the game. I am delighted we are still in the hat, I probably would have taken that at the start of the game.
"I thought it was a magnificent double save, really brilliant. I think if we had been beaten there I would have been really disappointed. I wouldn't have been sure we deserved to be beaten.
"I think on reflection a draw was probably a fair result."
Everton felt they were the more enterprising side and were frustrated at being forced to settle for a draw.
"We piled on the pressure and had opportunity after opportunity, culminating in that double save, which was tremendous from their perspective.
"The lads are disappointed in there that we couldn't turn pressure into a goal.
"It took us a while to get a foothold but the last 20 minutes of the first half and all the second half, I thought we were terrific and unfortunate not to win."
Bardsley opened the scoring with a long-range drive from a short free-kick after 12 minutes but the game might have been different had Drenthe been awarded a spot-kick four minutes earlier.
The Dutchman went down under a challenge from Craig Gardner but referee Andre Marriner saw nothing untoward. Replays, however, suggested Drenthe might have been unlucky not to get a decision.
Round said: "It was a clear foul. Maybe it was too early for the referee but he was right there on the spot and we don't know why he didn't give it.
"I thought Roy (Drenthe) didn't get much of the rub of the green. I thought he was fouled on numerous occasions.
"Whether they think he is diving or not - obviously he isn't because he is getting fouled.
"He is so small, he hits the deck. We are a little disappointed he didn't get more fouls than we expected."
Everton appealed for another penalty when Jelavic headed on to John O'Shea's arm but Round did not argue about that one.
He said: "That is a bit of an interpretation. We think it would be, they wouldn't, but the referee made a call and we have to accept that."