Lee McCulloch's second-half goal from the penalty spot secured the points for the Irn-Bru Third Division leaders, who could be crowned champions next Saturday if results go their way.
However, McCoist revealed his players endured a nervy 90 minutes yesterday.
"There were always concerns for us," he said.
"I felt that although we were reasonably comfortable after the sending off (of Ian Black), it's sometimes natural to defend a bit deeper and invite balls into your box.
"But we saw it out fairly comfortably and I'm delighted.
"Hopefully we can get the job done now as soon as possible. It's been a tough old season and a bumpy road, but all we said at the start of the season with this group of players was we wanted the opportunity to win promotion and win the league.
"We've given ourselves an incredible chance to do that now."
On the result, McCoist said: I thought it was a really hard-fought game as we expected it to be but we deserved the three points.
"The performance was just what was required after the disappointment of last weekend (when they lost 2-1 at home to Annan).
"I can't remember Neilly (goalkeeper Neil Alexander) having too much to do and we had a couple of really good chances in the first 15 minutes in particular."
Rangers captain McCulloch agreed with his boss.
"I think we were disappointed at half-time that we were not in front because we had a few half-chances, but thankfully we found a winner in the second half," he said.
"Once we got the goal I felt we sort of killed the game and it meant we won the game and it took us three points closer to where we want to be."
Referring to the spot-kick won by team-mate Lee Wallace, McCulloch said: "I felt it was a definite penalty.
"Lee is never one to go down easy and I spoke to him about it and he definitely felt contact."