They will play group stage football for the first time in a decade, despite being beaten by FC Zurich in the playoffs.
They should have been level on aggregate after trailing by two in the first leg in Switzerland.
They were dealt a blow early in the second-half whenJorge Grant was shown a second yellow card for diving in the box, which led to the winner for the Swiss champion.
The reward for finishing third in the Scottish Premiership last season was the chance to play in the Europa Conference League.
Europe's third-tier competition is still something to enjoy for Hearts, but they were masters of their own downfall at an initially loud Tynecastle.
The Edinburgh side responded well with a number of first-half chances, but were hesitant in the final third.
Liam Boyce missed a chance when he leaped up at the back post and headed down into the ground with the goal gaping, but the ball spurted over the bar.
Forrest had a great chance after a great move, but took too long to get a shot away as the defence recovered to block his low shot.
On 54 minutes, Grant went down early to anticipate a challenge that came later than he was expecting, to pick up a second caution, and Hearts' chances of maintaining their momentum were shattered.
After some good moments from the home side, the belief was drained from them and they were left with no chance of getting back into the game.
The first half was great. McKay and Forrest were a real threat and they were relentless without the ball.
They were the masters of their own downfall, and the lesson they need to take into the Conference League is to be more clinical.
Grant cost his team with a rush of blood in the box after they conceded two soft goals in the first leg.
His first yellow card for a tackle was harsh, but knowing he was on a yellow and going down in the way he did was stupid.
They can park this result and look forward to Friday's Conference League draw, where they are guaranteed six games.
It was a lesson learned and a chance to improve for the next test, with many of the names in the draw.
It's back to league action for Hearts as they host St Johnstone in the Scottish Premiership on Sunday, but before that they will have to wait for the group stage draw on Friday.
Formation 4-2--3-1.
2Smith3Kingsley15Rowles19CochraneThe formation is 3-4-1-2.
19Boranijasevic17Conde8Selnæs3Guerrero
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