Hibernian 4-3 Falkirk (aet)
Hibs staged one of the great Scottish Cup fightbacks to beat Falkirk and reach the final in a seven-goal epic.
The Bairns swept into a three-goal first-half lead thanks to Craig Sibbald's strike, Jay Fulton's stooping header and Blair Alston's finish.
But goals from Alex Harris, Leigh Griffiths, after his penalty was saved, and Eoin Doyle brought extra-time.
And Griffiths struck a beautiful 20-yard winner as time and Falkirk's energy and resistance ran out.
It took Hibs into their second Scottish Cup final in a row and keeps alive the Edinburgh side's hopes of ending that infamous 111-year wait to lift the trophy.
First Division Falkirk, in their first match under new manager Gary Holt, showed they have youthful promise but ultimately were unable to cope with the greater power and experience of their top-flight opponents.
Early defensive naivety allowed Griffiths a sight of goal only for his powerful low effort to be parried clear by goalkeeper Michael McGovern.
Fresh from an escape that was especially lucky considering the Scotland striker's 23-goal record ahead of the semi-final, the Bairns sped up to the other end of Hampden to take the lead.
Lyle Taylor has more than matched Griffiths' scoring rate since arriving from Bournemouth in the summer, but the 27-goal striker showed he can set them up too with a precise cut back that Sibbald dispatched clinically low into the corner of the net.
Scott Robertson, the Scotland midfielder whose confidence appears to have been shot since an ill-fated spell with Blackpool, mis-cued in front of goal as another good Hibs chance was squandered.
Falkirk were ripping apart the Hibs rearguard almost at will and, when Conor McGrandles flicked on a Sibbald free-kick flighted to the back post, Fulton stooped to head the second from eight yards.
The closest Hibs came to a reply was thanks to a miscued slice of a volley from Harris that struck the roof of the bar with McGovern scrambling.
A trickle became a flood as unhappy Hibs fans streamed from the national stadium after Falkirk sped into a three-goal lead.
Alston set Taylor clear and, when the striker's shot came off the legs of goalkeeper Ben Williams, the midfielder followed up to fire into the empty net.
Hibs boss Pat Fenlon hauled off the troubled Robertson, bringing on Danny Handling, but there were few signs of a change in fortunes as Taylor again got in behind the defence only to once more find Williams' outstretched pin.
All that changed after a break during which Hibs introduced Doyle for Paul Cairney.
McGovern brilliantly turned a Harris drive on to a post and Hibs were denied a clearcut penalty when Stewart Murdoch felled Tim Clancy.
The pressure paid off when a Harris drive from 25 yards skimmed McGovern's fingertips and nestled in the top corner.
It was now a case of old-fashioned shootie-in on an increasingly crowded Falkirk goal, with Griffiths belying his reputation by heading over when unmarked six yards out.
Having tempted fate earlier, Murdoch did concede a penalty after clearly clipping Handling.
However, McGovern produced a superb double save, first from Griffiths' low spot kick and then from Doyle's follow up.
Griffiths was only to be momentarily denied as he swept home a low Harris cross from 12 yards.
The energy was visibly sapping from the Bairns' legs and nearly all resistance was drained by the low Doyle drive that fired into their hearts and the back of the net from 18 yards.
Falkirk held on bravely in extra-time until Griffiths' almost inevitable strike five minutes before the end.
Taylor had a great chance moments later, but Williams' point-blank save prevented the tie going to a penalty shoot-out.