The Ibrox side needed two fine saves from their goalkeeper, Cammy Bell, to keep the scoreline blank at the beginning of the second half and McCoist concedes they benefited from a stroke of good fortune when a Lewis Macleod shot with 12 minutes remaining took a vicious deflection off Will Vaulks to spin into the net.
Nicky Clark, introduced early in the second 45 as an off-form Kenny Miller left the field with a hamstring problem, made it 2-0 and gave Rangers their first points of the SPFL Championship campaign following a home defeat to Hearts on the opening day.
"The performance was not as good as we had hoped," confessed McCoist. "I don't think we have started playing as well as we can, but, more than any time, the result was the most important thing as we had to get back to winning ways after the disappointing result against Hearts. I can't put my finger on it. We have good players and can, and will, do better.
"The intensity of the game probably caught everybody by surprise, but it is important we stay solid as a unit when we are not playing well and we did that okay. Our centre-backs could have done better, but, in saying that, they had a couple of important headers."
Peter Houston, the Falkirk manager, believed the hosts had been worthy of at least a point and could not detect a great deal of difference between his side and Rangers.
"If Rangers are going to be one of the better teams in the league, we are not that far away," he said. "For 78 minutes, I thought we were the better team. Our players put a lot into the game and deserved something from it."