Griffiths latched onto strike partner Garry O'Connor's classy through-ball to slip the ball home with five minutes left in Saturday's semi-final.
O'Connor had given Hibs an early lead but Rory Fallon levelled just before the hour mark with a stunning volley.
Aberdeen had been in the ascendancy after a toothless first-half display but Hibs regained the upper hand in the closing stages and their front pair proved too much to handle for the Dons defence.
The victory gives Hibs the chance to put aside a disappointing season in the Scottish Premier League season by lifting the trophy for the first time since 1902.
Aberdeen will be left to rue a shocking start as they remain waiting for their first cup final appearance since 2000.
Both sides were forced into changes with Jorge Claros coming into the Hibs midfield for the suspended Lewis Stevenson while Aberdeen midfielder Kari Arnason came in for the cup-tied Gavin Rae.
Fallon, who scored a double against Motherwell in the last eight, replaced Darren Mackie in the starting line-up.
The three SPL games between the teams this season have yielded just one goal and that was from a penalty that subsequently earned Aberdeen winger Peter Pawlett a two-match ban for diving.
But it took less than three minutes for O'Connor to open the scoring with the Aberdeen defence posted missing.
O'Connor first of all had all the time he needed in the box to pick up Isaiah Osbourne's blocked shot and lay the ball back for Griffiths, whose effort was quickly charged down by Mark Reynolds.
The ball fell for Hibs left-back Pa Kujabi, who aimed his cross for O'Connor, who had still not been picked up, and the striker guided the ball inside Jason Brown's near post as Andrew Considine made a belated effort to block.
Hibs comfortably repelled Aberdeen's initial attempts to hit back, which mainly centred around aerial balls.
Aberdeen began to get the ball down with Stephen Hughes becoming more involved in midfield, but Hibernian quickly closed them down inside their own half and sat deep to good effect when necessary.
The Dons could not carve a chance before the break and the lively Griffiths twice came reasonably close from long range.
Aberdeen brought on Fraser Fyvie for the start of the second half with Chris Clark moving from left to right midfield and taking Ryan Jack off.
The Dons got their first effort on goal in the 51st minute when Considine met Hughes' free-kick and headed straight at Graham Stack.
Kujabi shot wide from 25 yards after the Aberdeen defence scrambled to prevent several of his team-mates getting a shot on target and Fyvie soon volleyed well over from a difficult chance.
The match had opened up completely and Aberdeen followed a spell of pressure with a spectacular equaliser from Fallon in the 59th minute.
The move started with Reynolds, whose long ball looking for Fyvie's run was headed out by Matt Doherty. Fallon chested the ball down 25 yards out before sending a left-footed volley looping over Stack and into the top corner.
Aberdeen had the momentum and Fyvie curled wide after being given too much room 20 yards out.
Hibs lost Stack in the 73rd minute after he appeared to suffer a hip strain while kicking the ball and Mark Brown took his place.
Brown's first save was relatively comfortable, getting down to hold Mackie's effort after the striker had turned inside Kujabi.
Griffiths hit the outside of the post from a tight angle after linking with O'Connor on a counter-attack and the pair combined again to put Hibs in front in 85th minute.
O'Connor latched on to McPake's strong header from near the halfway line and played a first-time pass in behind Reynolds and Clark Robertson.
Griffiths was quickly on to it and the on-loan Wolves striker kept his composure to slip a right-footed shot past Brown and into the corner of the net.
Arnason shot not far over from 30 yards and Hibs survived a penalty claim when the Icelander's cross appeared to striker the arm of Kujabi.
Brown easily held Vernon's back-header during five minutes of stoppage time and Hibs held on to reach the final on May 19 against the winners of Sunday's clash between Hearts and Celtic.