Jack calmly slotted the ball into the back of the visitors’ net with 18 minutes to go, from Mark Reynolds’s cut-back, much to the delight of a big Pittodrie crowd still on a high from last weekend’s cup success.
Kris Boyd had fired Killie into an 11th-minute lead, swivelling and shooting past Jamie Langfield to stun the Dons.
However, Celtic Park penalty shoot-out ace, Adam Rooney, equalised six minutes later by rifling home from the edge of the penalty area to settle the home supporters’ nerves after a pre-match Scottish League Cup presentation, ahead of Jack’s winner.
A raucous Pittodrie was silenced when Boyd took Michael Gardyne’s pass in his stride and, with his back to goal, turned his man before angling a low shot across Jamie Langfield and into the net.
The Dons could have hit back immediately, but Cammy Smith’s left-foot shot from the edge of the area was well wide.
Aberdeen were soon on level terms when Smith’s pass found Rooney on the edge of the area, and the Irishman’s low shot found the back of the net (17), although Craig Samson in the Kilmarnock goal will surely feel he could have done more.
Niall McGinn missed a couple of chances for the hosts before the break, one a wickedly swerving shot passing narrowly wide of the target, and the visitors also came close through Vitalijs Maksimenko, but his header was comfortably dealt with by Langfield.
Boyd’s long-range effort, which failed to trouble Aberdeen keeper Langfield, halted a subdued opening to the second period and Dons substitute Barry Robson’s vicious inswinging corners saw Samson flap aimlessly at the ball.
Aberdeen won the match, after 72 minutes, following Willo Flood’s free-kick which was not dealt with by the Killie defence, and Jack popped up to stroke through a ruck of bodies with his weaker left foot from 15 yards from Reynolds’s assist.
It should have been completely over as a contest when Jack’s pass found McGinn on the edge of the area with 10 minutes remaining, but the striker passed up his third clear opportunity of the afternoon, firing wide of the upright.