The Boys in Green had their backs to the wall almost throughout but were able to claim a creditable result, arguably swinging the tie in their favour ahead of the return fixture in Dublin three days from now.
Pione Sisto wasted Denmark's best chance in a dominant first half but, despite seeing more than three-quarters of the ball, there was to be no way through for the Scandinavian outfit at the Parken Stadium.
Next to come close for the Red-White was Christian Eriksen, who struck the ball well from 25 yards out but sent it too central for Randolph to routinely keep out.
Nicolai Jorgensen and Sisto also squandered first-half chances, the latter getting on the end of a parried Eriksen shot but failing to keep the follow-up attempt on target.
Ireland's only opening of any note during the opening 45 minutes fell the way of Cyrus Christie, who almost lofted the ball over Kasper Schmeichel late in the half, before Jeff Hendrick's shot was eventually blocked aside.
Despite the defensive display, the visitors - aiming to reach their first World Cup finals since 2002 - were doing all that was required to keep the opposition out and take a draw into the return fixture.
They managed to achieve that by remaining strong at the back for the remainder, restricting the Danes to minimal clear-cut chances.
Denmark finished the game on top, though, coming close to snatching a winner through substitute Yussuf Poulsen and Larsen, the former of whom had his headed attempt from six yards out turned over the bar by Randolph.