The Saints were in a good position to progress midway through the second half with the tie evenly poised at 1-1 on aggregate and the visitors having been reduced to 10 men after Karen Muradyan was shown a second yellow card.
However, a long ball from Ararat Arakelyan completely caught out the St Jonhstone defence and Norayr Gyozalyan raced clear to lob goalkeeper Alan Mannus, giving the visitors a vital away goal.
Brad McKay’s late strike made it 2-2 on aggregate, but time ran out as the hosts pressed for a decisive third goal and they bowed out on away goals.
“We can play better than that but we were in control of the game at 1-0 and with them down to 10 men,” said Wright.
“I don’t think Alan Mannus has had one serious save to make and we shot ourselves in the foot by giving away a really soft goal again. That is two really disappointing goals we have given away against them.
“Once you give the away goal away with 15 minutes and injury time to go, it was always going to be difficult to get to three goals. We got one, we just couldn’t get that other one. We are extremely disappointed that we were in control of the game and gave a soft goal away at that stage.
“It was one ball and it shouldn’t be that easy at any level at football. That has cost us the game and the tie. We don’t often give away bad goals like that.”
Wright was also unhappy the referee missed an elbow on defender Steven Anderson, who had to be substituted with a cheekbone injury moments before Alashkert’s equaliser.
He said: “The referee has missed an absolute assault on Steven Anderson. I have just watched it back and he actually gave the free-kick the other way and booked Ando, who is now away to hospital with possible cheekbone damage, a swollen eye and concussion.
“It is quite serious. He has been vomiting – it is a bad one. If anyone had done that in the street they would be lifted, it is a blatant elbow to the face.
“But I don’t want to sound as if I’m bleating. The referee didn’t cost us the tie, we cost ourselves the tie over the two legs and the players are extremely disappointed.”