The holders lost to an injury-time winner by Mark Millar after Farid El Allagui had grabbed a double for the Irn-Bru First Division side and Rangers fought back through goals from Dorin Goian and Nikica Jelavic.

Jelavic - an injury doubt beforehand - and Steven Naismith both started on the bench and were only thrown into the action when the SPL side were trailing in the third-round tie.

McCoist had refused to take the credit for Sunday's Old Firm triumph but held his hand up on this occasion.

He said: "Hindsight is a great thing. The Jelavic thing was taken out of our hands to a certain degree because we were told he wasn't really fit to start and play for 90 minutes, although that might have been the case if we had gone to extra time.

"But I pick the team and I shoulder all the responsibility for the result. It's as simple as that.

"I'm not going to say what might have happened if I'd picked him or picked someone else because we will never know the answer to that.

"What we do know the answer to is that I fully take the responsibility for the team."

McCoist admits his side simply failed to match the efforts they showed against Celtic at the weekend.

He said: "I'm bitterly, bitterly disappointed at the way we played for an hour. We had to play at a high tempo like we did previously and we didn't do it.

"It was as though someone turned on a switch when we lost the second goal and we realised we were in danger of being knocked out of the tournament.

"The difference was incredible but effectively it was too late. You can say we came back into the game at 2-2 but Falkirk fought for everything and gave it everything they had.

"We were miles short and that's really disappointing."

Asked if fatigue or complacency was a factor, the Rangers boss said: "There certainly wasn't tiredness and I would hope there wasn't complacency.

"There is a reason why we didn't play at a tempo for an hour and that's what we will have to look at as it was a massive problem for us."

Falkirk boss Steven Pressley had nothing but praise for his players.

He said: "I feel exceptionally proud of those young players. Whatever credit they get over the next couple of days, they certainly merit.

"They showed desire and focus to win the game and they deserve all the praise going. Nobody could have asked for more of them.

"I feared the worst if the game had gone to extra time but at 2-2 I still thought we had a chance.

"They played without fear but they played with their minds as well. They remained focused and the mentality was brilliant."