Griffiths ran onto a pass from Garry O'Connor in the 85th minute and beat Jason Brown with an exquisite low finish.
Hibernian went ahead within three minutes as O'Connor notched his 15th goal of the season but Rory Fallon cancelled out his strike with a stunning second-half volley.
Hibs will face either Celtic or Hearts who meet in the other semi-final.
Aberdeen manager Craig Brown made two changes from the side which defeated Dundee United last weekend, with the cup-tied Gavin Rae missing out and Darren Mackie dropping to the bench. Kari Arnason and Fallon came in. The fit-again Russell Anderson had to settle for a place on the bench, with Andrew Considine and Mark Reynolds continuing in central defence.
Lewis Stevenson missed the match for Hibs due to suspension, with Honduran midfielder Jorge Claros stepping in. Paul Fenlon opted for a 4-4-2 formation with O'Connor and Griffiths leading the line up front.
Hibernian came out of the blocks quicker and took an early lead when O'Connor found himself unmarked to stab home a Pa Saikou Kujabi cross from the left. The left-back was given far too much time to cross the ball, while the Aberdeen marking in the middle left much to be desired.
Aberdeen grew into the game but their final ball was all too often inaccurate. Hibs were dominant in the air, with the likes of James McPake excelling in central defence.
Hibs were pressing superbly and the front pairing of O'Connor and Griffiths looked far more dangerous than Scott Vervon and Fallon at the other end.
Chances were at premium in the first-half, although the lively Griffths fired two powerful shots just wide of Jason Brown's left-hand post.
The half-time whistle was greeted with boos from sections of the Aberdeen support.
Graham Stack held onto an Andrew Considine header as Aberdeen showed some early intent in the second-half.
Brown's half-time team-talk obviously had a positive effect as the Dons dominated for long spells of the second-half.
Chances remained few and far between, although Kujabi shot wide after some excellent work from Ivan Sproule.
Aberdeen levelled the match in style with a stunning dipping volley from Fallon. Matt Doherty attempted to head away a long ball, but it fell to Fallon who controlled and volleyed sublimely over Stack from well outside the area. The volley flew over Stack and then dipped under the crossbar.
Griffiths was a lively presence all afternoon and he ran onto a pass from O'Connor down the right and shot against the outside of the post as Hibernian finished the match stronger.
They got the goal their performance just about merited in the 85th minute. The impressive McPake won another header at the back for Hibs, and the ball was steered on by O'Connor to his strike partner Griffiths who escaped his marker and fired clinically into the bottom left corner.
Hibernian reverted to a 4-5-1 formation as they soaked up some late Aberdeen pressure to secure their place in the final.