The away side made the better start of the two teams and took the lead through Leigh Griffiths after 15 minutes when he struck past the goalkeeper from inside the area.
Kilmarnock pushed forward to get themselves back into the game and came close from a corner, which bounced around the box before the away side just managed to clear.
Hibs restored their one-goal advantage on 38 minutes through Matt Done, who tapped into the bottom left corner from a Griffiths assist after some poor defending from the home side.
Ross Barbour threatened early in the second half, cutting inside and hitting his shot just wide of the post.
Paul Heffernan then had the chance to bring the scores level, but was bundled over in the area, giving the home side a penalty.
Heffernan stepped up and buried the spot kick to make it 2-2 on 72 minutes.
Griffiths burst through the Kilmarnock defence soon after and almost put his side back in front, but Cameron Bell saved well to deny the striker.
Griffiths did give Hibs the lead on 82 minutes, getting on the end of a long ball and knocking it past the goalkeeper.
Barbour was then shown a red card on 87 minutes for pulling back Griffiths who was in on goal.
The away side were awarded a penalty for a handball on 89 minutes and Griffiths stepped up to secure a hat-trick and his side's semi-final place.