The Black Cats head into Wednesday night's fourth-round replay at Middlesbrough having earned themselves a replay courtesy of returning hero Fraizer Campbell's equaliser at the end of last month.
Arsenal, who head for the Stadium of Light in the Premier League on Saturday, await the winners in a home tie the following weekend, although O'Neill will not even contemplate a quickfire return to Wearside for the Gunners until it is secured.
He said: "I would be really surprised if a Sunderland supporter who really knew about the game would take anything for granted now, particularly for the kind of run we have had in recent years.
"I don't think that for a second. That's certainly not my attitude. We have a very, very difficult game that can go either way, and I wouldn't even look at Arsenal at all.
"Obviously, we have got Arsenal at the weekend, and that's a different issue altogether in terms of the league.
"What I can say is that we will just give everything to try to win the game - but that won't automatically give us victory at all."
Sunderland's run of form since O'Neill's arrival in December has been little short of remarkable, with a return of 22 league points from a possible 30 easing them well clear of the relegation zone.
However, Boro are one of only four sides to deny them victory during that period - Tottenham, Everton and Chelsea are the others - and for that reason, the Ulsterman knows they will have a fight on their hands once again at the Riverside Stadium.
He said: "The only enjoyment we all get, either as players of supporters, would be in the winning of the game.
"But I am sorry, I cannot guarantee that at all and I think it will be very, very difficult for us."
O'Neill will not be able to call upon loan signings Wayne Bridge and Sotirios Kyrgiakos, who arrived after the registration deadline, while Wes Brown, David Vaughan, Lee Cattermole, Matt Kilgallon, Titus Bramble and Nicklas Bendtner remain on the injured list.
However, he could turn once again to Campbell, who he chose to rest for Saturday's 1-0 league win at Stoke following his return from a long-term knee injury.
O'Neill said with a smile: "I listened to the medical team - perhaps too much. In fairness, if most medical teams had their way, the players would never play!
"But I have to be guided by them. He's been out a fairly lengthy time and the last thing I would want to do is set him back.
"He could get a setback anywhere, but the lad is full of adrenaline at the moment. He is flying and obviously we are delighted that he's come back so well."
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