Leigh Griffifths sealed an extraordinary victory when he fired home a brilliant 20-year effort past Michael McGovern with five minutes of extra-time left.
Falkirk had struck after just six minutes when Blair Alston robbed Kevin Thomson and slipped a pass to Lyle Taylor, who cut the ball back to the 18-yard line for Craig Sibbald to place his shot into the bottom corner.
Sibbald started the move that led to the second when his free-kick was nodded on by Dods for Conor McGrandles to hook across the box where Jay Fulton (18) swooped to head the ball home from six yards.
The Bairns struck again when Fulton robbed Paul Cairney 40 yards from goal - the ball broke in front of Taylor and his low drive was parried by Ben Williams straight into the path of the alert Alston (30) to tap into the net.
Taylor was denied a fourth just before the break as he raced on to a long pass and strode into the Hibs box, but Williams blocked the striker's shot with his foot in what would turn out to be the defining moment.
In the early stages of the second-half McGovern parried an Alex Harris volley against the post after the midfielder exchanged passes with Eoin Doyle and the Falkirk keeper tipped the follow up shot from Griffiths over the bar.
Hibs were denied a strong claim for a penalty when Stewart Murdoch scythed down Tim Clancy from the resulting corner, right in front of the assistant referee.
But a minute later McGovern could not stop a 25-yard drive from Harris spin into the net with a slight deflection off Darren Dods - despite the keeper getting his gloves on the ball - 3-1.
McGovern saved a weak shot from Danny Handling after great play from Harris before Doyle and Griffith both put efforts over the bar from inside the penalty area.
And McGovern denied Hibs again when Murdoch pulled down Handling (72) to concede a penalty, this time awarded, which the Barins stopper saved from Griffiths and also the rebound from Doyle.
But Griffiths then crashed home from close range after Falkirk were opened up by Ryan McGivern and Harris, with 12 minutes remaining - 3-2.
And Griffiths then flicked on to Doyle to run at a back-peddling Dods and blast his low shot past McGovern from 20 yards to level the tie five minutes later.
Griffiths' extra-time bullet sent Hibs through to the final where they will play either Celtic or Dundee United who play on Sunday.