Beau Reus' brilliant penalty shootout saves secured 10-man Hearts Conference League progress as they fought back to defeat Rapid Vienna after an enthralling play-off second leg.

Calvin Miller's spectacular extra-time volley put the visitors into the lead, before Nenad Cvetkovic met Moulaye Haidara's cross at the back post to nod in a late sucker punch, sending the game to spot-kicks.

Pierre Landry Kabore had forced the additional 30 minutes for Wouter Vrancken's side when he chested in an equaliser to cancel out Rapid's opener from Ercan Kara. By the time Kabore netted, Tomas Magnusson had been shown a second yellow card with 15 minutes of regulation time to go.

After their remarkable title bid fell just short last season, finishing runners-up and securing a chance at Champions League qualification, Hearts lost ties in the top flight of European competition and Europa League before looking unlikely to advance to the Conference League proper multiple times.

Vrancken's team were level on aggregate after a 2-2 first-leg draw at Tynecastle but the home side looked the superior outfit from kick-off.

They routinely peppered the Hearts goal and pinned them inside their box for swathes of the first half as the visitors survived multiple scares.

A bobbling ball was eventually cleared from inside the six-yard box and Petter Nosakhare Dahl's whipped cross was not tapped over the line as the home side remained relentless.

As Hearts' heads drifted towards getting in level at the interval, Romeo Amane swiped at goal from the edge of the area and Kara was unmarked in the six-yard box to convert after a deflection diverted fortuitously into his path.

Claudio Braga stole into the area, picking up on a slack pass, but his shot was met by Rapid goalkeeper Niklas Hedl before a rasping Oisin McEntee strike was dealt with similarly.

Hearts almost made a spectacular start to the second half when Blair Spittal's sweetly struck volley fizzed narrowly wide of the post. Their tactical tweaks at the interval was allowing them more time on the ball and a greater influence on the game.

James Wilson, who scored a spectacular long-range strike in the first leg, broke in behind and saw the ball deflect just wide of the post as he tried to cut back across a defender.

Braga's poked attempt was thumped clear off the line from the resulting corner as the visitors grew even more into the contest.

But, just moments after Magnusson had an incredible opportunity to nod Hearts level again, the Icelandic midfielder was shown a second yellow card for a late challenge on Nikolaus Wurmbrand.

Their attacking was not to be diminished, though, and an instinctive, if not unorthodox, finish from Kabore sent the tie to extra time.

Hemmed in their own defensive third, Hearts broke forward through another substitute, Rogers Mato, who found Miller with a deep, looping cross from the left that the winger fired in with spectacular technique, and appearing to injure his hamstring in the process.

Hearts still had 20 minutes to see out and Cvetkovic's late finish from close range was a deflating blow after such a heroic effort to rally back.

Miller, Jordi Altena, Mato, and Sabah Kerjota all scored from the spot before Oisin McEntee missed with Hearts' first concrete chance of progression.

But Rapid's Moulaye Haidara had his shot saved from the spot before former Celtic attacker Marco Tilio's was also met by the diving Reus to secure a famous win for the team from Tynecastle.

And they will play six more matches in the Conference League this autumn, with the draw for the league phase taking place on Friday.

After 45 minutes, it seemed inevitable Hearts had run their European race - Rapid were resilient and looked the most likely team to score next.

From the second-half kick-off, Vrancken's side were transformed. The tactical tweaks were not dramatic, the game plan was not overhauled, but the attitude, the determination, the hard work ramped up.

As regulation and extra time ticked on, it was almost comical the extent to which Hearts were hurling themselves in front of the ball to stop it crossing the line.

At one point, Miller could barely leave the centre circle but stuck to it and set the tone with the first spot-kick.

Every single player played their part, in what was by far the best collective performance of the season and easily comparable with some of the club's greatest results of this century.

Hearts have had a rebuild of sorts this summer after a plethora of high-profile departures.

This performance is proof they are well on the right track.

Hearts head coach Wouter Vrancken: "Amazing, amazing. When you see the game, we played a really bad first half, a little bit similar as we did in Benfica.

"We made agreements, we had a game plan, and we didn't follow up, we didn't show the guts to follow up.

"When you play against a side like Rapid Vienna, when you don't show it, they can play through you. We were already pretty angry at half-time, and we came out, we did much better.

"We took the game in our hands, we were dominant at that moment. This is already a mental strength, coming back on 1-0 down and coming back like this.

"Again, in the extra time, you try to take the game with you to Hearts and to Edinburgh, then you get a mental kick again in the last minute of the extra time.

"Again, I'm so proud of these guys that they stay in the game, focused and concentrated to win the penalties."