The Boys in Green were on the back foot for the majority of the game at the Parken Stadium and they had goalkeeper Darren Randolph to thank for keeping them on level terms ahead of Tuesday's second-leg in Dublin.
O'Neill was certainly pleased with the battling spirit his side showed, as they avoided defeat in an away qualifier for the sixth successive match, but he acknowledged they will have to play better in the return leg if they are to secure their place at next summer's finals in Russia.
"I thought it was a tough evening for us and the players did splendidly. They put a lot of effort in and it's all to play for now," he told Sky Sports. "We have to win a game. And I think we have to score a couple of goals because Denmark are capable of scoring too.
"We didn't score a goal obviously, but we're still in the tie."
Ireland are looking to qualify for their second successive major tournament, but they have not reached the finals of a World Cup since 2002.