0923 GMT June 23, 2018
O'Neill's side had just 28 percent of possession in Copenhagen but was well-organized and disciplined as it frustrated the home side, BBC reported.
"It was a tough evening and the players did splendidly," said O'Neill.
"We've come out the other side and it's all to play for. We'll have to win a game and we might have to score a couple because Denmark are capable of scoring."
The Republic of Ireland is aiming to make its first appearance at a World Cup finals since Japan and Korea in 2002, when Mick McCarthy's side reached the last 16.