Dundee Utd 3-4 Celtic (aet)
Celtic will meet Hibernian in the Scottish Cup final after beating Dundee United in the second seven-goal Hampden thriller within 24 hours.
Kris Commons gave Celtic an 83-second lead, but Gary Mackay-Steven levelled before Jon Daly headed United in front.
Victor Wanyama headed an immediate reply and Commons chested in to put Celtic ahead after the break.
Another Daly header took the game into extra-time, during which substitute Anthony Stokes nodded in the winner.
It not only keeps alive Celtic's hopes of a league and cup double but ended a three-visit losing run at Hampden.
United can take heart from a performance that suggests they are becoming a thoughtful and elegant football team in recently-appointed manager Jackie McNamara's own mould.
The question was not so much how United would cope without a prolific goalscorer like Johnny Russell, whose return from a broken ankle came just too late for the striker to even appear on the bench.
United had, after all, gone eight games without defeat during his absence.
It was more how the formation adopted to compensate would handle a Celtic side for which Scotland wingers Commons and James Forrest were passed fit to start.
The answer appeared to come after just over a minute of play when, with United's three-man central midfield dragged to the right, Commons exploited the space to sidestep the desperately backtracking John Rankin and Willo Flood before blistering a 20-yard drive past goalkeeper Radoslaw Cierzniak.
Thoughts of their 6-2 thumping at the hands of Celtic in their last meeting must have crossed the United players' minds at that point, but they came back with admirable tenacity and no little skill.
Efe Ambrose blocked a Mark Millar drive off the line after Celtic goalkeeper Fraser Forster conceded an indirect free-kick inside the penalty box and Kelvin Wilson had to clear with a diving header from in front of his own goal.
United's defence was looking fragile to cross balls and Mikael Lustig should have extended Celtic's lead instead of sending a close-range header off the inside of a post after being found by a precise Emilio Izaguirre delivery.
Just as United looked like capitulating under the pressure, Daly found space after Wilson unwisely lunged for a desperate Barry Douglas clearance.
Mackay-Steven and Ryan Gauld seemed to take an age over their little triangle around the exposed Ambrose, but the former eventually curled the ball past Forster.
A fired-up United had a good claim for a penalty turned down when Wanyama barged into the back of Mackay-Steven.
United's second was not long in coming, though, as Daly bowed to flick home an inswinging Douglas free-kick.
Mackay-Steven and the goal provider appeared to switch off at the restart as the former allowed Izaguirre to cross and Wanyama rose handsomely above Douglas to level from eight yards.
United recovered to end the first half slightly on top, but - as with Hibs against Falkirk in the first semi-final - the team in green were the ones all go after the break.
Izaguirre was again the provider as the ball took a fortunate bounce off the chest of the unmarked Commons and in off the post.
Commons was then denied a hat-trick by Cierzniak's foot at the front post.
However, just as it looked like Celtic would kill the game, Mackay-Steven's cross found the head of Daly and the striker's looping header sneaked in off the far upright.
United survived a late pounding of their goal to take the game into extra-time.
A Daly header came off the post and Michael Gardyne's follow up was cleared off the line by Forster.
Forrest was denied by the goalkeeper's legs, but the winger's next act was to provide the cross from which Stokes powered in the winner.
Celtic's greater class and experience edged the game, but the real winner was football itself after two brilliant last-four ties that blew away the doom and gloom that has been surrounding the Scottish game.