The Danes had won 5-1 in Dublin in last year's World Cup play-offs but, with Christian Eriksen absent, both sides looked to be lacking a cutting edge as they battled to a point.
Ireland boss O'Neill was content with how the game played out and praised the hosts' work with the ball, yet he acknowledged their failure to trouble Kasper Schmeichel, who enjoyed a quiet night in the visiting goal.
"Early on, I thought we allowed Denmark too much possession," O'Neill told Sky Sports. "They grew into the game, whereas we were sitting back a little bit.
"But when we got the ball, we pushed forward with some purpose. We have to start creating a few more chances. Cyrus [Christie] has had a great shot. That said, I don't think Denmark caused us too many problems."
However, with Ireland offering little in attack, O'Neill was pleased to at least stop the rot at the back after a thrashing in Wales in their group opener.
"In terms of the Wales game and going back over a year, I think a clean sheet was very important," he said. "It's us rebuilding again, getting that confidence again. Get a clean sheet and you're in the game."
James McClean similarly hoped that this result could be used as a springboard, with Ireland two points behind Wales ahead of welcoming Ryan Giggs' men to the Aviva Stadium.
"It wasn't pretty, but we dug in. We battled very well and showed a good attitude," McClean said. "It's the first point on the board and it's better than it was against Wales. This is progress.
"It's a clean sheet - I don't think we've kept one since Denmark away [in the play-offs]. It was nice to pick up a clean sheet. The lads defended excellently; now it's just about having that cutting edge and putting the ball in the net.
"Hopefully, on Tuesday, we can match the hard work and determination with some goals."