Prior to the penalty round, both Croatia and Denmark were locked at 1-1 after neither of them found the net after 90 minutes and in the extra time.
At the beginning of the match, defender Mathias Jorgensen gave Denmark an early lead in the first minute, when midfielder Thomas Delaney clipped the ball to the former inside the penalty box, thereby completing the goal by clipping it past Subasic.
However, three minutes later, Croatia equalised when striker Mario Mandzukic took advantage of a botched clearance, wherein the ball bounced past Danish defender Andreas Christensen's head, giving Mandzukic the perfect time to guide the ball past goalkeeper Kasper Schmeichel.
Thereafter, Croatia and Denmark fought frantically to give their respective teams a dominating position.
In the 27th minute, Subasic denied Denmark's second goal when forward Martin Braithwaite played a shot from the right side of the penalty box, hitting straight to the Croatian goalkeeper, who just dabbled the ball.
Two minutes later, midfielder Ivan Rakitic attempted a long-range shot but was thwarted by Schmeichel.
By half-time, both Croatia and Denmark were deadlocked at 1-1.
The two countries lobbied hard to score but were unable to break past the defence displayed by their respective teams.
Midfielder Luka Modric attempted several times to find the net but failed to do so every time. This also included a penalty in the extra-time, wherein he played a straight shot, but was beautifully deflected by Schmeichel.
With the two teams failing to create opportunities, the match entered into the penalty shootout round.
There were nerves in the camps of both Denmark and Croatia. After all, it was just only one team who could progress to the round of eight.
Danish midfielder Christian Eriksen took the first penalty, but his shot hit the goalpost, inviting celebrations from the Croatian camp.
However, Croatian midfielder Milan Badelj too missed out an opportunity to score in the penalty as Schmeichel made a miraculous save.
The brief deadlock ended when Denmark's Simon Kjær and Croatia's Andrej Kramaric finally opened their respective teams account by scoring a goal.
While Danish midfielder Michael Krohn-Dehli scored his team's second successive goal, Modric took his revenge and hit a straight shot towards the net. With this, both Croatia and Denmark were locked at 2-2.
The penalty shootout suddenly took a dramatic turn when Subasic made two spectacular saves to block Lasse Schone and Nicolai Jorgensen, putting Denmark in a spot of bother.
In the end, Rakitic ensured that there were no hiccups in the round by signing off with a beautiful curl shot to put Croatia in the quarterfinals for the first time since the 1998 World Cup in France.
Croatia will now lock horns with Russia, who earlier shocked 2010 champions Spain in penalties, in the quarter-finals at the Fisht Stadium in Sochi on July 7.