The veteran was replaced by Nicky Clark at the start of the additional 30 minutes.
The 36-year-old gave Mark Warburton's men the lead with a great finish after 16 minutes at Hampden Park but with Hoops defender Erik Sviatchenko equalising just after the break, an extra half-hour was needed to try and separate the Old Firm rivals.
But Miller's part in the game was over by that point as Warburton replaced him with Nicky Clark for extra-time meaning that he had to watch Tom Rogic cancel out Barrie McKay's stunning effort, that had put the Scottish Championship side back in front, and send the game to penalties.
Luckily for Miller, the Light Blues came out on top with Rogic missing the decisive spot-kick. And after the victory was confirmed, the forward explained how difficult it was to watch from the sidelines.
"For me it was a fantastic day - I have never really felt like that before. I was close to tears. Watching from the bench was torture, just torture. You are hitting every ball, every header, every clearance," he told The Scottish Sun.
"We got there in the end and I've always been one for celebrating and being happy.
"But this was emotional for me. It was just a great game to be involved in."
After experiencing how tense the game can be from the bench, Miller will be hoping that Warburton keeps faith with him for the upcoming final with Hibernian, and that his side can end the game inside 90 minutes to make the roller coaster of emotion slightly less dramatic.