It proved to be a match to forget for the Saints strikers, as Michael O'Halloran missed a sitter and substitute Steven MacLean saw red for two silly challenges.
Aberdeen manager Derek McInnes made only one change from the side that beat Celtic in midweek, with Barry Robson dropping to the bench in place of Jack, who returned from his long-term injury.
And it was the Scotland Under 21 midfielder who fired the Dons in front after eight minutes with a 25-yard effort that somehow managed to find the net.
Jack received the ball from Peter Pawlett on the edge of the penalty area and then beat defender Steven Anderson, before seeing his weak shot trickle under St Johnstone goalkeeper Alan Mannus.
After a succession of Aberdeen corners, St Johnstone tried to hit the hosts on the counter-attack and came close to equalising 10 minutes later when forward Lee Croft picked up the ball on the right-hand channel and cut inside Andrew Considine, but his shot was hit straight at Jamie Langfield.
The home side remained in control throughout the first half, dominating possession in the midfield, and they almost doubled their advantage when Adam Rooney was given enough time on the edge of the box to weigh up a shot, but this time Mannus was equal to it.
Aberdeen's flowing passing moves in midfield were creating problems for the Saints defence, and good link-up play down both channels created chances for Johnny Hayes and Willo Flood, but both failed to trouble Mannus.
St Johnstone finished the first half the better side and passed up the best opportunity to equalise six minutes before the interval.
The Aberdeen defence failed to deal with a loose ball in the box, which Stevie May chased down before picking out O'Halloran. The Irishman found himself six yards out and unmarked but could only squander the chance wide of Langfield's left-hand post.
St Johnstone started the second half the better team and had a penalty appeal waved away after May felt he was barged over by Alan Tate on the edge of the penalty box, but referee Willie Collum deemed the challenge to be a fair one.
The visitors went even closer on 66 minutes when O'Halloran raced clear but could only angle his shot against the inside of the post, with the Aberdeen defence able to clear the ball to safety.
St Johnstone's unlucky afternoon was completed when MacLean was sent off for two mistimed challenges, the first on Barry Robson moments after coming on as a substitute, with the second coming a minute from time after bundling over Nicky Low.
The visitors continued to try press for the equaliser and tried to make the most of their possession, but the Dons defence were able to match the attacks as they held out for the win and another clean sheet.