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O'Neill recalled that Celtic finished without a trophy when he led his side to the Uefa Cup final in 2003 and yet fans still regarded it as "a great season".
However, he said: "You could say that in recent years the fans have maybe been spoiled with no real challenges in many aspects and been able to coast home in the last six or seven games of the season.
"Here, Hearts have arrived, maybe a few years before they thought they would do. Rangers have come strongly and spent some money in getting good players in.
"It has been a challenge all season and certainly been a challenge in my couple of spells in here when you're winning some matches and you're still trying to peg teams back.
"But we're in with a chance anyway - that's the point."
Celtic came from two goals down to draw with Rangers in the league on Sunday and went on to overtake their city rivals in the table after beating Aberdeen on Wednesday.
"We've had three games in six days, which has been a tall order for us, all away from home," O'Neill said. "We withstood that, which was a great effort.
"Because of that, we were able to withstand the game at Pittodrie. That was never going to be easy, but it was one we had to win to stay in the competition.
"If we play like we did in the first half [at Ibrox], we shouldn't turn up, but the second half gave us great confidence.
"It is testament to the players' character as much as anything else."
O'Neill is not sure how much the quarter-final result will affect the league title race.
"Naturally, taking the game on its own, it would be a big boost for us getting to the semi-final," he added.
"Will it eventually have a bearing on how the league will pan out? I don't think it will have a serious bearing, but in terms of confidence building, of course it matters.
"There are tough matches ahead for us, tough matches ahead for the other sides concerned, so I really don't know.
"I suppose if you get beaten in the game, it's how quickly you can get over it."
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