Until halftime, Rapid were clearly the superior side, but they let it slip in the dying minutes despite being a man up.
SK Rapid Wien misses out on the Conference League group stage following a 1-2 home victory against Hearts of Midlothian.
Rapid started the match better and created the first chances.
Following a superb through ball, Adamsen tested away keeper Reus from a tight angle, but the shot was saved (10').
Rapid remained the dominant side thereafter, but initially struggled to break down the deep-sitting Scots.
In the 34th minute, Rapid finally got their reward: Amane played in the onrushing Demir, who squared the ball across the face of goal. The pass took a slight deflection and fortuitously reached Kara, who had an easy tap-in at the far post.
Hearts only threatened before the break with two long-range shots, both saved by Hedl.
After the break, the Scots looked significantly improved.
Following a corner, Spittal had a shot from just outside the box, but sent the ball narrowly wide of the left post. The best chance then came again from a corner, but Kara cleared off the line (59').
Rapid first threatened in the second half through Dahl. After combining with Demir, his shot went wide of the goal (63').
Hearts continued to pose a threat, particularly from set pieces. In the 75th minute, they narrowly missed the equalizer when a header sailed just wide of the goal.
Just a minute later, Hearts' Magnússon was shown a red card for a foul on Wurmbrand (76'). However, the Scots were undeterred: From a corner, Kaboré stood completely unmarked and bundled the ball over the line with his chest to make it 1-1 (79').
In the closing stages, Rapid had two more chances to win the match before extra time.
First, a shot from Wurmbrand went just wide of the goal (85'). Shortly after, Raux-Yao was denied by the post in the dying moments (87').
In extra time, Rapid remained on the front foot, but were hit with a clinical counter-attack. Miller scored with an overhead kick, sending the ball past a diving Hedl, to give Hearts a 2-1 lead (99').
Heart-stopping finale in Hütteldorf
In a desperate late push, Demir came closest to an equalizer with a long-range shot (109'). Just when it seemed all hope was lost, Rapid struck back: Cvetković connected with a low cross from Haidara in the 119th minute to make it 2-2.
In the penalty shootout, Miller confidently converted Hearts' first attempt. Kara also held his nerve from the spot.
After the Scots also converted their next two penalties, Schaub scored for Rapid. Haidara then became the first Rapid player to miss from the spot.
Demir converted as Rapid's fourth penalty taker. Hedl then saved the Scots' first match point, but Tilio missed Rapid's fifth and decisive penalty. Rapid thus failed in the playoff and missed out on European football this season.
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