The Hoops were leading 1-0 when defender Meekings appeared to stop a close-range goal-bound header from Leigh Griffiths with his hand but the official behind the goal, Alan Muir, and referee Steven McLean waved away penalty appeals.
The decision left Celtic boss Ronny Deila and his players fuming as Craig Gordon was later sent off for tripping Marley Watkins to concede a penalty, which Greg Tansey scored.
David Raven grabbed the extra-time winner for the Highlanders after John Guidetti had cancelled out Inverness striker Edward Ofere’s low drive.
However, Meekings felt Caley were rewarded for their ambition after lining up in a 4-4-2 formation and taking the game to Celtic.
He said: “It has come off my arm, but he is two yards away from me. I have just gone to throw my body in the way. Unfortunately it has come off my hand but fortunately it hasn’t been given.
“There was a long way to go and we still had a lot to do. It’s always going to be difficult against Celtic so when you get decisions like that it’s a bonus.
“I thought we played really well, we passed it really well and pressed them a lot better than we did last week when we played them.”
Meekings, whose side will play Falkirk in the final, added: “It’s a massive achievement for all of us. It’s something that the club has never done before and for it to be this bunch of boys to have taken us there is massive.
“It’s down to us. We know we believe in each other to go on and win the final. We have to make sure we are ready to go one better.”