It looked like being another long 90 minutes for Derek McInnes’ men when Anthony Stokes netted his third goal in four days against them, after just 42 seconds.
But Saints soon levelled through the recalled Gavin Swankie before goalkeeper Alan Main saved Derek Riordan’s penalty.
The visitors deservedly sealed a quarter-final place late on thanks to goals from substitute Chris Millar and superb strike from Jody Morris.
Hibs boss John Hughes kept faith with the side from Saturday’s Scottish Premier League win.
Injuries forced opposite number McInnes into four changes, including first starts of the season for midfielder Swankie and defender Steven Anderson, while Kenny Deuchar and Steven Milne were recalled up front.
Stopping Stokes would have been high on Saints’ agenda after his double against them on Saturday.
But it took less than a minute for them to fall victim to the striker again.
Darren McCormack cut onto his left foot before curling a 20-yard strike Main could only parry to Stokes, who kept his cool to fire into the net.
But Saints were level six minutes later, finally finding the finishing that had eluded them at the weekend.
Paul Hanlon sliced his clearance from a left-wing cross straight to Swankie, who hit a lovely low strike across Graham Stack from the edge of the box.
The visitors began to dominate: Liam Craig appealed for a penalty after being held by McCormack, while Milne charged down an attempted Stack clearance minutes before failing to connect cleanly with an overhead kick.
Hibs were simply unable to handle Saints’ pressing game and they should have fallen behind on the half hour when sensational skill from Milne saw him spin into the box.
The striker’s finish was too close to Stack, who was also on hand to thwart Deuchar’s follow-up.
Two minutes later, referee Charlie Richmond courted controversy when he decided Graham Gartland’s shoulder-to-shoulder challenge on Liam Miller warranted a penalty.
Riordan stepped up but saw a spot-kick saved for the second time this season, Main springing superbly to his right.
Danny Grainger was the first player booked for chopping down McCormack, who then came to Stack’s rescue after the goalkeeper fumbled a looping cross.
Saints had the first sniff after the break, Ian Murray’s slip letting in Swankie, whose cross evaded everyone.
Anderson was booked for a crude tackle on Riordan on the edge of the box before Morris followed suit for a foul on Miller.
Hibs waited less than 13 minutes before bringing on Merouane Zemmama for McCormack in a bid to wrest control of midfield.
Saints responded by withdrawing the excellent Swankie for former Hibs midfielder Filipe Morais, who missed Saturday’s game through illness.
Main needed his defence to put the ball behind after parrying a Stokes cross and the goalkeeper then kept out Miller’s strike from the resulting corner.
Danny Galbraith came on for the ineffective John Rankin before Zemmama played in Stokes, whose shot was brilliantly blocked behind by Grainger.
David Mackay was cautioned for a late touchline tackle on Galbraith before Saints threw on Millar for Craig.
It was an inspired substitution, the midfielder fed brilliantly by a slide-rule pass from fellow replacement Morais before slotting the ball past Stack.
Millar had been on the field for two minutes.
Stuart McCaffrey came on for Anderson before the outstanding Morris put the seal on the comeback with eight minutes remaining.
Hibs failed to clear a corner but Morris’ first-time 20-yard half-volley was right out of the top drawer.