Villa came from behind and deserved the share of the spoils handed them by Everton central defender Phil Jagielka's last gasp own goal.
O'Neill said: "I thought before the game that we probably had to win the whole lot of the matches that were left to give us a proper chance (of the Champions League).
"It would have been a tall order and I would have thought that, certainly after this result, to give ourselves any chance we'll have to win all five.
"It's not impossible but it will be tough. You'd need a bit of luck here and there but we'll go to Portsmouth on Sunday and we'll give it everything."
O'Neill had nothing but praise for the resilience shown by his side after their FA Cup semi-final heartbreak against Chelsea, even if the set-piece defending for Tim Cahill's two goals left a lot to be desired.
He said: "It was tough. I had a chance to watch the game between Chelsea and Bolton on Tuesday and I thought Chelsea were a wee bit sluggish-looking, maybe from their exertions against us on Saturday.
"So, from our view point, I thought we were terrific against Everton. I thought we kept going.
"I thought the fitness levels were incredible seeing as it was the same players in action. I thought our resilience was absolutely top class.
"Maybe a point doesn't suit either side at the end of the day in terms of our aspirations, for Everton in terms of European football, and for ourselves for maybe something higher but it was a great game."
Everton manager David Moyes is desperate to return to Europe and does not believe it would be a distraction for his squad.
He said: "We will keep going and see how it goes. I think Villa were a long way ahead of us at one stage so I've got to give the players credit for getting themselves back in a strong position.
"We will keep trying to see what we can do but, if you had said to me in October we'd be in this position now, I'd have said you were kidding me on.
"The players have done a great job. We've had two defeats in 20 games. I think the galling thing is we've had three draws in a row where we could have done with picking up another point or two."
Moyes added: "Europe is worth fighting for. I think it is important for our football club to be there.
"There might be some managers who might think 'miss out on it because it will give you a better chance in the league'. At the moment that isn't my thought.
"My thought is to make Europe because I think our football club can carry it. I really do. If anyone doesn't want it, I'll take their place.
"I think our football club needs it as a development. I have to try and get there and, also with attracting players, I think it can be important as well."
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