There was no need for extra-time on this occasion, though.
Jason Cummings fired the Edinburgh club ahead from the penalty spot only for Saints to equalise four minutes later through Joe Shaugnessy.
But man of the match John McGinn sent the 12,000 Hibs fans in the Gorgie ground wild as he drilled home what proved to be the winner 16 minutes from time and into a Hampden final against either Celtic or Ross County.
Hibs were dealt a blow before kick-off with on-loan Celtic striker Anthony Stokes ruled out with a back strain meaning just one change to the line-up from last week’s victory over St Mirren, Dylan McGeouch returning from injury to replace Marvin Bartley in midfield.
Bartley was on the bench where he was joined by new signings goalkeeper Otso Virtanen and, back for a third time at Easter Road, midfielder Kevin Thomson.
Alan Stubbs would have been pleased with the start his side made, Liam Henderson whipping in two early free-kicks which were well taken by Saints goalkeeper Alan Mannus.
Michael O’Halloran got the chance to display the blistering pace which has made him the Perth club’s main threat, but he was snuffed out by David Gray’s tackle.
But Hibs continued to take the game to their Premiership opponents and were rewarded with a 29th minute penalty as Henderson went down under the challenge of Chris Millar, the Saints players reacting with fury as referee Steven McLean immediately pointed to the spot.
Referee McLean’s namesake, Saints striker Steven MacLean was booked for his protests before Cummings stepped up to send Mannus the wrong way with a low shot to send the 12,000 or so fans packed inside Tynecastle ecstatic, his 18th goal of the season.
But their joy was to last just four minutes, Hibs unable to completely clear their lines following Simon Lappin’s free-kick as St Johnstone exerted some pressure, leaving former Easter Road midfielder David Wotherspoon clipping in a cross which Shaugnessy met with a superb header which soared into the top corner of goalkeeper Mark Oxley’s net.
It was the first goal Hibs had conceded in the competition having beaten Montrose, Stranraer, Aberdeen and Dundee United en route to this semi-final.
Hibs began the second half as brightly as they had the first but the game became rather scrappy, neither side able to retain possession and so create anything.
But a Lewis Stevenson foul on Murray Davidson inches outside the penalty area wide to Oxley’s left almost brought Saints a second, Lappin going for goal from the free-kick with a left-footed shot which clipped the top of the cross bar.
Hibs responded, Stevenson’s ball finding Chris Dagnall who found his way to goal blocked, the ball falling for Cummings who attempted a shot with the outside of the his post forcing Mannus into a save before Dagnall fired in a low effort on the turn which left the Saints goalkeeper beaten but the ball clipped the outside of his post and went wide.
Mannus as in the right place, though, as Dagnall swivelled on a Henderson corner, finding himself too close to the goalkeeper to capitalise on the half-chance.
The game had burst into life, Gray blocking Steven Anderson’s shot before Hibs broke in numbers, Henderson electing to go alone only to drag his shot wide.
But a second for Hibs was merely delayed, McGinn used his strength to get on the ball before dispatching a superb low shot which zipped into the bottom corner of Mannus net from 20 yards.
Dagnall had the chance to wrap it up for the Capital side, his smart run met by an equally clever chip from McGinn but in attempting to go for the far corner he was inches off target. McGinn then charged into the Saints area only to run up a cul de sac, leaving Saints still with some hope.
Oxley through himself to his left to push aside another Shaugnessy header from Lappin’s corner and O’Halloran blazed over as the clock ticked into four minutes of added-on time and Saints threw everything forward.
Hibs: Oxley, Gray, Hanlon, Fontaine, Stevenson, Fyvie, McGeouch, McGinn, Henderson (McGregor 89), Dagnall, Cummings (Keatings 86).
Substitutes not used: Virtanen, Bartley, Boyle, Gunnarsson, K Thomson.
St Johnstone: Mannus, Shaugnessy, Mackay, Anderson, Easton (Scobbie 46), Millar (Craig 71), Lappin, O’Halloran, Davidson, Wotherspoon (Sutton 78), MacLean.
Substitutes not used: Clark, Cummins, Fisher, C Thomson.
Referee: Steven McLean.