However, the Saints looked keen to defy the odds and would have levelled up the tie on 17 minutes if not for the athleticism of Dobrivoj Rusov, who tipped Stevie May's goal-bound effort past the post.
Unperturbed, the Scots continued in the same vein and saw their perseverance rewarded five minutes before the break, as May struck from outside the area to put his team within one goal of progression.
St Johnstone had goalkeeper Alan Mannus to thank for keeping it that way after he produced a smart save to keep out a header on 69 minutes.
Wright's troops began to tire as the half grew older and their bid to reach the fourth and final round suffered a knockout blow on 81 minutes.
Gary Miller played the ball back to Mannus, but unwittingly headed into the path of Martin Mikovic, who made the defender pay the ultimate price by slotting past the keeper to end St Johnstone's European odyssey.