It was The Danes who started brighter in Kadriorg, located just to the north east of Tallinn, creating some good chances in the early exchanges. However, as the first-half wore on Kai Nyyssönen's side began to come into the game.
Teemu Jäntti was twice denied by Danish keeper Mads Hermansen, Enoch Banza seeing a shot blocked in between, as Finland created the better chances in the opening 45 minutes.
Jaakko, who was making his 14th international appearance, was involved in the move which should have seen The Finns take the lead eight minutes after the break; somehow Jyri Kiurulle missed the target after Jaakko's corner found him completely unmarked in the middle.
Tuure Mäntynen and Jäntti went close again shortly after and Finland's cause was helped further with 20 minutes to play as substitute Nikolai Frederiksen was shown a red card. The Finns, however, couldn't turn their man advantage into a goal as they were forced to settle for an opening day draw.
The other game saw Italy, who are next up for Finland, cruise past Estonia 3-0 to take control of Group Two. That game will be played on Saturday, kick-off 1pm UK time.