The Jambos lost the second leg in Gorgie 1-0 thanks to an 81st-minute strike from Florian Rohner, securing a 3-1 aggregate triumph for Franco Foda’s men.
Hearts dominated the first half without scoring and paid the price on 51 minutes when Jorge Grant was sent off for a second bookable offence for simulation in the Zurich penalty box.
Despite the defeat, Neilson hailed his players for the performance they put in in front of a packed Tynecastle.
"I’m very proud of the players,” said Neilson. “I thought they were outstanding. I’m proud of the club. They’ve had a long journey – from administration through to promotion. The whole objective of the club was to give back to the fans. There’s a lot of pride. For the first 60 minutes, I thought we were outstanding. I thought we were the better team. We took on the Swiss champions and put them under massive pressure. The sending off changes the game and made it very difficult. There is a disappointment, but overall, pride in the players, the club, the fans and the board.
Neilson was at odds with Belgian referee Lawrence Visser over the red card decision for Grant, saying: “When I speak to our players I say to them, when you’re in the penalty area, you can’t come off your feet. If you do, you have to make contact with the ball. He comes off his feet. The striker, in my opinion, is under no obligation to get out the way. So as soon as there’s contact, it should be a penalty. But the referee has seen it a different way. And it’s cost us the tie.
“At that point we are still 2-1 down from the previous game. They get a chance to then open up If you give these players time on the ball, they are good players, they are internationals, the league champions. We found it hard then to get pressure when we needed it.
“He has made his decision. Everyone has a different opinion. People say there is contact, people say he goes down too easily, but if you are a defender and you come off your feet and you make a challenge and you do not hit the ball, and you make contact with the player, it’s a penalty. But we didn’t get it.”
Hearts now drop into the Europa Conference League, with the draw for that competition taking place on Friday afternoon. “It gives us massive confidence,” Neilson said of that tournament and what it could do for belief for the rest of the year. “We’ve shown we can go away from home and play well and we can play well at Tynecastle with massive intensity and put teams under massive pressure. We should have won the game but it didn’t happen.”