A John Baird goal four minutes from the end of extra time consigned Rangers to defeat in the final at Easter Road last Sunday.
McCoist and his players were criticised after photographs emerged of the manager singing karaoke at a post-match function.
But, in an interview on the club's official website, McCoist said he and his side felt the cup defeat just as much as the fans.
"I can assure everybody the players were feeling it along with the staff and all the supporters," he said.
"It has never been any different at this club. You win and lose together as a team and a squad and everybody feels the same.
"We were extremely low after Sunday but it is our job to get everybody back up and focused again."
McCoist and his players have an immediate chance to redeem themselves when they play Dundee United at Ibrox in the Scottish Cup semi-finals on Saturday.
McCoist said: "Rangers teams of the past have lost big, big games and they've always bounced back and showed a great mental attitude to do that. This team will hopefully be the same.
"We've seen and we can tell the players are well up for it and they're really motivated and focused on getting the job done.
"We're still really disappointed as you'd expect but we can't sit back and feel sorry for ourselves.
"There's absolutely no doubt we can move on. It's a game against a team which will definitely be favourites against us and they've had some good performances this season.
"Dundee United have been scoring goals for fun and they've got some very good players from the middle to front.
"At the same time, our lads know what is expected of them and the game is getting played at our stadium.
"We'll have a fantastic support there and in an ideal world we'll give them something to shout about."
Meanwhile, Rangers goalkeeper Cammy Bell could be set to miss the clash with Dundee United due to injury - according to his understudy Steve Simonsen.
Simonsen tweeted: "Feel so sorry for Cammy, it's Rotten luck, but I will step up; I want this so much, for me, for my teammates, my kids, & The Rangers Family!
"I hasten to add i'm not saying Cammy is 100% out of the game. Just that I'm fired right up if it's the case! This is why I'm here, to step up."
Watch Rangers take on Dundee United from 12.30pm this Saturday live on Sky Sports 2