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 in a crowd of just under 24,000.
Memories of infamous Scottish Cup exits to Berwick Rangers in 1967 and Hamilton in 1987 began to permeate the thinking of the Gers fans as the second half unfolded, but the home side drew level in the 77th minute when Bilel Mohsni put pressure on Neil Parry from David Templeton's cross just under the bar, with the ball ending up in the net.
The two teams will meet again in the replay at New Douglas Park on a date yet to be decided, with the winners meeting Dundee United in the semi-final at Ibrox, while Aberdeen also take on St Johnstone at the Govan stadium. The last-four ties will be played on Saturday, April 12 and Sunday, April 13.
Gers boss McCoist said: "I am obviously relieved 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.
Rangers: Bell, Foster, McCulloch, Mohsni, Wallace, Aird, Black, Law, Templeton, Shiels (Little 63), Daly (Faure 75). Subs not used: Peralta, Simonsen, Crawford, Cribari, Smith.
Albion Rovers: Parry, Reid, Michael Dunlop, Ross Dunlop, Russell (Flood 75), Ciaran Donnelly, Phillips, Chaplain, Cusack (Kennedy 69), McGuigan, Dallas (Crawford 21). Subs not used: Innes, McGinley, Tiffney, Walker.
Booked: Russell, Ciaran Donnelly, Parry, Flood.
Ref: John Beaton.