Robbie Brady's 85th-minute header saw Ireland clinch a 1-0 victory in Group E on Wednesday and sealing third spot in the process.
Wes Hoolahan had missed a glorious opportunity earlier before putting in the cross for Brady's goal.
Speaking after the game, the Ireland boss praised his players for overcoming the disappointment of not being awarded a penalty in the first half when James McClean appeared to be fouled.
"There wasn't a player out there in an Irish shirt who didn't perform heroically," O'Neill said. "There was real disappointment not to be given the penalty, but we had to get over it.
"We actually took some heart from that."
The former Celtic manager felt that Ireland's chance may have gone when Hoolahan failed to convert in a one-on-one with Salvatore Sirigu.
He said: "I must admit, I was concerned that we wouldn't create as a good a chance again [as his miss].
"But we did, he put in a lovely ball and Brady's gone in as bravely as anything."
While an exhausted O'Neill praised his side's spirit, he said they had shown their quality at the tournament.
"We dominated for incredible periods and played some phenomenal stuff," he said. "Obviously there's a great spirit among the players, but that won't win games alone.
"Some players have come of age and done brilliantly in the last two years.
"But we keep going, the players know, they never give up."
"I look forward to the game. We're playing the host nation and they are very strong," he said.
"They'll be strong favourites to beat us but if we can recover in time then we want to play as strongly as we did tonight."