Villa created a host of first-half chances on Wednesday night but needed a late own goal from Everton defender Phil Jagielka to earn them a share of the spoils.
It means they have now drawn six of their past seven matches at Villa Park and their hopes of a top-four finish are now hanging by a slender thread.
The result also did no favours to Everton in their bid to claim a Europa League spot.
O'Neill said: "Maybe a point did not suit either team in terms of our respective aspirations, Everton in terms of European football and ourselves in terms of something higher.
"I thought before the game to give ourselves a proper chance (of Champions League football) we have to win all six remaining games.
"Now I would say we have to win the whole lot from here. It is not impossible but it will be tough. We will just give it all that we can."
O'Neill felt the least his team deserved was a point with Gabriel Agbonlahor's 50th Villa goal and Jagielka's helping hand cancelling out two goals from Tim Cahill.
He said: "It was a great spectacle from start to finish. We deserved the equaliser at the end.
"We should have been in front at half time. We had to chase the game all evening.
"We got our first equaliser and then conceded another poor goal from our viewpoint but we again came back."
Everton boss David Moyes was in a similar mood to O'Neill in refusing to write off his side's Europa League hopes even though they still trailed Villa by four points and have played a game more.
He said: "We will keep going. We will keep trying. Europe is important and we will keep fighting to try and get in there.
"When you consider where we were in October, the players have done a great job and we have lost only two of 20 matches.
"It is a bit galling that the last three matches have been draws and we could have done with an extra point or two.
"I thought we were going to get over the line against Villa but we didn't quite manage it.
"At 2-1 up, we had a couple of great chances to seal the game and didn't take them."
Moyes felt Villa striker John Carew was offside immediately before Jagielka turned Ashley Young's cross past his own keeper Tim Howard.
He said: "When Young crosses the ball, Carew is just offside but it would have been a hard one for the referee to spot.
"I thought it was hard on our keeper because he made a couple of great first-half saves. He is disappointed he didn't keep out the second goal but his overall performance was magnificent."