The Scottish side were 1-0 up from the first leg but conceded a 75th-minute equaliser and, after extra-time failed to produce a winner, the Belarusian visitors won the penalty shoot-out 3-2.
St Johnstone had not been defeated at home in Europe for 42 years but they had an early escape when Raman Vasilyuk slipped past Frazer Wright after 10 minutes, only to be foiled by a Steven Anderson block.
The hosts then had a pair of great chances to extend their aggregate league but squandered them both.
Firstly, as the half-hour approached, Nigel Hasselbaink produced a lovely pass to play in Stevie May, but the striker mishit his shot and Vladzimir Bushma got down to save.
May then returned the favour by cutting the ball back for Hasselbaink, whose half-volley was just wide of the target.
The hosts wasted some free-kicks in decent positions after the break and Aliaksandr Sviarchynski then put a free-kick just over for the visitors, as the nerves increased for the home fans.
And their worst fears were realised when Milnos Rnic let fly with a first-time volley from the angle of the box which got past Alan Mannus' despairing dive.
There was plenty of drama in the extra period, Sviarchynski shooting into the side netting from close range after Tam Scobbie's slip while Steven MacLean was foiled by Bushma after the change of ends.
The home side then thought they should have had a penalty for a foul on Rory Fallon, before Maclean fired wide.
The shoot-out began terribly for St Johnstone, Dave MacKay and Paddy Cregg missing their first two attempts.
However, Mannus then saved consecutive efforts by Vital Kibuk and Siarhei Kazeka, giving MacLean the chance to level at 3-3 with the 10th kick of the shoot-out.
But Bushma dived to his left to save and knock out the Scots, and Minsk then had Siarhei Sasnouski sent off after the final whistle for goading the dejected hosts.