Ciaran Donnelly shot the visitors into a shock lead in the 13th minute after the Light Blues had failed to deal with a corner and the rest of the match was played out amid growing frustration within the home supporters, until Bilel Mohsni equalised in the 77th minute when he bundled the ball and Rovers keeper Neil Parry over the line in his determination to get to a David Templeton cross.
The two teams will meet again in the replay at New Douglas Park On Tuesday week with the winners getting Dundee United in the semi-final at Ibrox, with Aberdeen taking on St Johnstone in the other last-four game.
Gers boss McCoist said; " I am obviously relived that we are still in the cup. For 70-odd minutes we weren't going to be in the cup.
"But we got the break, got the goal and the main feeling is relief.
"I thought Albion Rovers defended very well but on pressure we certainly deserved to take something out of the game and the draw was what we got.
"I have had a look at the draw but as you saw today, it would be wild of me to have the audacity to look forward to a home tie against Dundee United. I am certainly not going to do that."
The Rovers bench complained about the leveller which required Parry to have treatment but McCoist insisted that there was nothing controversial about referee John Beaton's decision.
"It is a goal," said the former Rangers striker. "If you look at Mohsni, not having seen it again, he jumps and I don't think he has any idea where the goalkeeper is and it hits his head and goes in.
"It's a goal for me.
"I think the easy decision for the referee was to give a foul but I thought he was very brave in giving the right decision."