In a pulsating match at Rugby Park, Griffiths opened the scoring in the 15th minute only for James Dayton to equalise.
Matt Done restored Hibs' lead in the 39th minute before Paul Heffernan slammed in a penalty to level again. But Griffiths was not to be denied as he notched twice more, the second from the spot, with Killie ending the match with 10 men after defender Ross Barbour was sent off late on for a tug on the on-loan Wolves player at the edge of the box.
Hibs, who famously have not won the tournament since 1902, will join Celtic, Dundee United and First Division Falkirk in the draw at Hampden and it is mostly down to Griffiths, who celebrated his first professional hat-trick.
Both teams to time to settle in a scrappy start to the match but in the 10th minute Griffiths took the initiative with a speculative shot from 25 yards only to see it skid wide of the target.
However, it was no surprise that the prolific Hibs striker was on target when he was presented with a better chance moments later.
Killie defender Jeroen Tesselaar's clearance only reached the middle of the park where David Wotherspoon headed it back to send Griffiths clear on goal. With the former St Mirren defender giving chase, Griffiths kept his composure before knocking the ball low past Cameron Bell, whom he had beaten for a late equaliser in the SPL game during the week.
Killie could hardly increase their commitment and work-rate but struggled to find any rhythm against the buoyant visitors.
However, out of the blue the home side found themselves level when William Gros' powerful snap-shot from the edge of the box, after being set up by Heffernan, was parried by Hibs keeper Ben Williams who was then helpless as Dayton raced in to lift the ball into the roof of the net.
The threat of Griffiths was ever-present as both sides remained locked in combat, mostly in the middle of the park, but Killie were architects of their own downfall at Hibs' second goal.
Bell rolled the ball out to Mohamadou Sissoko but when it was returned rather robustly from 15 yards away he failed to bring it under control, allowing Griffiths to pounce.
When Bell felled the Scotland cap in an attempt to recover the situation, referee John Beaton allowed play to go in for a second or two to allow Done to hammer the ball into the empty net from eight yards for his first goal since his loan move from Barnsley in January.
Killie dusted themselves down and battled back and two minutes from the break Heffernan lift the ball over Ryan McGivern's head before flashing a powerful volley just past Williams' far post.
Striker Cillian Sheridan replaced defender Rory McKeown as Shiels reshuffled his side and in the 71st minute Kilmarnock were awarded a penalty.
Left-back Lewis Stevenson tangled with Heffernan in the box as the Killie striker tried to get on to the loose ball following Williams' save from Dayton. Contact looked minimal and indeed referee Beaton seemed to hesitate before pointing to the spot, but the Irishman kept his composure to hammer the ball past the helpless Hibs keeper.
The visitors roared back and Griffiths forced a good save from Bell with his angled drive but there was more drama when the two men tangled after Griffiths had chased a long James McPake pass through the centre, pursued by Sissoko.
The keeper appeared to have the ball in his hand, while Hibs were shouting for a penalty - but amid it all, Griffiths scooped the ball into the net to the jubilation of the Hibs fans behind the goal.
There was more excitement to come.
Barbour was red-carded for a last-man tackle on Griffiths, who then drove in a penalty to secure the match ball after Gros had used his arm to block the resultant free-kick, and with that, the Easter Road men moved one step closer to their Holy Grail.
- Sports & Recreation