Fraser Aird came off the bench to grab the only goal of the game in the 92nd minute to seal a 10th win in a row for the Irn-Bru Third Division leaders.
But despite acknowledging the outcome was hard on Gardner Speirs' men, McCoist's overriding feeling was one of relief.
He said: "Of course I have sympathy. Gardner is a good lad. His team gives him absolutely everything and I'm sure you will agree they were really well organised.
"They probably had their better opportunities when we were a bit reckless in pushing forward.
"It would need to be a cruel man who didn't have any degree of sympathy for the opposition.
"At the same time, I've got to look after our own corner and I'm thrilled with the victory. It was a relief.
"The longer the game goes on, you wonder if you are going to get a goal or possibly even lose one.
"To be honest, I think we lost our shape a little bit a couple of times late on in pursuit of a goal, which is inexcusable.
"I can understand the desire to get a goal but we should have been more solid.
"I thought it was a goal worthy of winning any game. It came at the perfect time for us."
Rangers' last visit to the national stadium saw Nikica Jelavic grab an extra-time winner to defeat Celtic in the Scottish Communities League Cup final in March 2011.
The difference between that game and today's league clash could not have been more stark, following a summer of turmoil that resulted in the Light Blues dropping to Scottish football's bottom tier.
McCoist said: "It's bizarre really but I think I probably celebrated the goal today as much as I did that one. It was a really hard-fought three points.
"It shows you the difference in the teams and it shows you how far we've got to get back to, when you look at the personnel that played in that particular game.
"But the boys who were representing Rangers today kept going.
"We can play better than we did today and it would be wrong of me not to say that. But it would be wrong of me not to remind everybody that's seven wins in December.
"I've got be harsh with them at times but when you take a step back, they've been fabulous this month."
The Spiders had four players booked, as well as the two yellow cards which resulted in James Brough's second-half dismissal.
But boss Speirs was more interested in praising his own players than discussing the performance of referee Willie Collum.
He said: "I think there was inconsistency in the decisions but that's not the reason we lost the game.
"I'd rather just focus on how well our guys did, how brave they were and how well they played."