The home side twice pegged their opponents back, but had no answer to the in-form Griffiths, who took his tally for the season to 22.
The striker, on loan from Wolves, put Hibs in front with 15 minutes of the quarter-final tie gone, latching on to a long header from David Wotherspoon and firing low past Cammy Bell from just inside the box.
The home side were back on level terms soon afterwards when Ben Williams could only palm away Willian Gros' pile-driver and James Dayton reacted first to smash the ball home.
The game looked to be heading into the interval all-square, but seven minutes before the break Hibs restored their lead following a mix-up in the Kilmarnock defence.
Mo Sissoko over-hit a back-pass to Bell, the keeper panicked under pressure from Griffiths and the ball bounced off the Hibs hitman into the path of Matt Done, who gratefully tucked it away for his first goal since arriving on loan from Barnsley.
Griffiths continued to terrorise the home defence in the second half but Hibs could not stretch their advantage and they were made to pay 18 minutes from time when Killie grabbed the equaliser from the penalty spot.
Paul Heffernan went down under a challenge from Lewis Stevenson and the striker picked himself up to fire the resultant penalty past Williams.
The game looked to be heading for a replay, but Griffiths had other ideas. First he went down under a challenge from Bell and kicked the ball from the keeper's grasp into the net.
And a few minutes later he slammed home a penalty, awarded after Gross had stopped his free-kick with an arm.
The original set piece was awarded after Griffiths had been brought down by Ross Barbour, which resulted in a red card for the the Kilmarnock defender.