Rangers spurned several good opportunities to score and they paid the price when John Baird bagged the winner for doughty Rovers three minutes from the end of extra-time.
The League One champions departed the fray to a chorus of boos, and manager McCoist told the club's website: "I'm extremely disappointed as you can imagine, as are the players.
"The game itself, we didn't play well but I thought we had the better chances.
"When you don't take them you obviously run the risk of losing a goal and that's the way it worked out.
"We didn't play well enough but we should still have scored given the number of chances we created.
"But there was a slackness about us on occasions defensively today and it came back to bite us.
"That was the first time we haven't scored this season in a game and it's our second defeat, albeit both of them have been in extra time.
"But it still hurts and I thought the manner in which we lost the goal was very, very poor as well.
"I've been here long enough as a player and a coach and a manager to know that there will be a lot of unhappy people, none more so than ourselves, the players and the staff.
"As down and as low as we are we can't stay down there because we've got a massive game coming up next week.
"We'll regroup and see how the injuries are in the next 24 hours and then we'll start to prepare for a massive semi-final."
McCoist confirmed that Lee Wallace, who was replaced after 65 minutes, has a thigh strain and is a doubt for next week's Scottish Cup last four date with Dundee United.
Rovers manager Grant Murray added: "When you get to this stage, it's all about winning the trophy.
"The boys have given me everything all season and to come to Easter Road and beat Rangers is absolutely outstanding. They are absolute heroes and they've got a gold medal for themselves.
"I am so proud of them. The fans were outnumbered today, but this is going to live with them for the rest of their lives."