Noel King's side were forced to play just under half the match with 10 men following the dismissal of Sean McGinty soon after the break.
McGinty brought down Denmark's Youssef Toutouh to receive his marching orders.
But even with a numerical disadvantage, Ireland managed to create chances to snatch victory on the break.
Blackburn defender Anthony O'Connor went close with a header on the hour mark while Aston Villa’s Samir Carruthers just failed to connect with a low Michael Harriman cross on 78 minutes.
At the other end, Irish keeper Sean McDermott impressed with two decent saves to foil Bassel Jradi late on and keep the hosts at bay.
Ireland begin their next Euro U21 Championship qualifying campaign against the Faroe Islands on August 14.