The hosts dominated the midfield for much of the game and created plenty of chances but had problems finding the target and it took the introduction of substitute Pierre Emile Hojbjerg in the 73rd minute to break the deadlock.
The strapping midfielder, who replaced Lasse Schone, was only on the pitch for four minutes when he timed his run into the box perfectly to send a header flying past goalkeeper Darren Randolph to give his side the lead.
The setback saw the Irish throw caution to the wind as they poured forward looking for an equaliser and got their reward when Duffy brought them level five minutes from time when he climbed highest to head home Alan Judge's free kick.
The draw left Ireland on seven points from three matches at the top of the five-team group, three points ahead of Switzerland who have a game in hand.
Earlier, Georgia beat visitors Gibraltar 3-0 to register their first points after three games and move into third place above Denmark, who have two points in fourth but a game in hand. Gibraltar are bottom without a point from their two matches.
(Editing by Ken Ferris)
