O'Neill's side had just 28% of possession in Copenhagen but were well-organized and frustrated as they frustrated the home side.
Denmark had chances but a combination of their poor finishing and impressive saves from Darren Randolph ensured O'Neill's side will start the second leg on level terms.
The visitors' only clear opportunity saw Kasper Schmeichel save Cyrus Christie's attempted chip before Jeff Hendrick's follow-up was blocked.
"It was a tough evening and the players did splendidly," O'Neill said. "We've got out for a side because Denmark is capable of scoring.
" They had one really great chance. Overall we restricted them. An away goal would have been great but we did not carve out that many chances. We need to be more creative [in Dublin]. We're still in the tie. "
The Republic of Ireland is aiming to make their appearance at a World Cup finals since Japan and Korea in 2002, when Mick McCarthy's side reached the last 16.
The second leg of the play-off for a place in Russia next summer is at the Aviva Stadium in Dublin on Tuesday.
Halfway there ...
Having claimed their play-off place in the last round of qualifying fixtures - courtesy of James McClean's solitary goal against Wales in Cardiff - the Republic are unlikely to take anything for granted in the second leg.
However, they have decent pedigree in these two-legged, last-chance routes to reach a major finals.
This is the eighth time they have contest a play-off to reach a major tournament - more than any other European nation - and they have prevailed in three of the last four.
The only losing effort in that run came at the hand of Thierry Henry, when the Arsenal legend infamously handled the ball to ensure France progressed to the 2010 World Cup.
However, there will be (19459008)
Despite their lack of attacking intent, Republic midfielder Harry Arter believes the result must be viewed as a good one.
"It feels a bit closer than it did before the game," the Bournemouth player told Sky Sports.
"Before the campaign, if we had a game against Denmark to qualify for the World Cup, we'd have to bitten off your hands for it.
" A draw has always got to be seen as a positive result. "19459008]" It's important not to go into the second tie losing. "
It's important not to go into the second tie losing."
The Republic's quest to reach The legend of a picked up booking in Denmark.
This included Arter, who (19459007]
"The referee handled the game really well - he was aware of who was on a booking," added Arter.
'Easy to read, difficult to beat'
However his assessment proved frustratingly accurate for the home side.
O'Neill pulled a minor surprise in his starting XI with his selection of Callum O'Dowda and not Glenn Whelan in midfield, which hinted at an ambition to pursue an away goal to defend in Dublin.
But, true to form, this was a stubborn, defense-minded display from (19459007)
For the majority of the game, the away side had five or more men strung across the width of the pitch near the edge of th eir own box, with the two biggest - Shane Duffy and Ciaran Clark - tasked with hammering clear, by any legal means, all that Denmark could throw at them.
It proved effective for the most part, especially in denying Denmark's star player Christian Eriksen seventh international game running.
The tactics predictably gave lone striker Daryl Murphy a thankless task. The Nottingham Forest forward touched the ball just 20 times, none of them in the home penalty area, before he was replaced by Shane Long with 25 minutes to go.
Keeper Randolph was, just as unsurprisingly, much more involved .
The highlight of his display was a superb double save from Jens Stryger Larsen before reacting quickly to block and gather the follow-up from Cardiff striker Andreas Cornelius.
He almost ruined his good work, though, with a poor push-out from Eriksen's shot but, after gathering the loose ball, Pione Sisto fired inches past the far post.
Man of the match - Darren Randolph (Republic of Ireland)
Republic's rare blank away
- The Republic of Ireland are unbeaten in each of their last four play-off qualifiers away from home (W1 D3) and have avoided defeat in all of their fixtures on the road in this World Cup qualifying campaign (W3 D3).
- Indeed, the Republic of Ireland has not lost any of the international meetings with Denmark (W2 D3), keeping clean sheets in each of the last three (W2 D1).
- This was the first time the Republic of Ireland has failed to score in an away international since November 2014; (19459016)
- Denmark has not lost a single international fixture in 2017, winning four and drawing four of their eight games and conceding just four goals - they have not suffered since October last year against Montenegro (1-0).
