Their reward is a tie against Denmark who finished second in Group E, behind Poland, and O'Neill is expecting a tough test, particularly with an in-form Eriksen in the team.
"Christian Eriksen is playing out of his skin at the moment, playing brilliantly for club and country. He's scored a lot of goals as well, very, very important goals for Denmark," he told reporters.
"You could go around the squad really, they are a very decent side, Denmark. We will have to tread carefully, be up for both games and take our chance.
"Eriksen is a top quality player, he's exceptional really, he's been in great, great form for not just club but country and playing brilliantly.
"They have very, very good players all around the pitch, playing in big leagues as well, and they are a formidable team."
The first leg of the tie will be played in Denmark on November 11, with the return leg back at the Aviva Stadium in Dublin on November 14.
Republic of Ireland went unbeaten away from home in the group stage, with wins in Moldova, Austria and Wales, and O'Neill is hoping they can take that form into the play-off.
"Well the second match is in Dublin and I hope that we still have something to play for by the time that we get there," he added.
"I think that all the teams that were unseeded, probably didn't want to play Italy, so there was a 50/50 chance of us getting them from that viewpoint, but that said it's going to be a difficult game for us but we'll take our chance.
"Our away record in this particular competition has been very strong, which is great news. I don't think the players fear that but they will be very wary."