McKay, along with team-mates Lewis Macleod and Robbie Crawford, vowed to pledge himself to Rangers until 2017 back in August.
However, Scottish Football Association rules barred the player from signing his new contract until he turned 18 on December 30.
With the red tape now out of the way, McCoist revealed his delight at securing the player on a long-term agreement.
He told the club's website: "We have now completed the contractual process with Barrie. He deserves it.
"We want the best years of his career to be at Rangers, as we do with all of the young players who are doing well for us.
"It's a boost for them to sign these deals and it's a boost for us too. It also shows the fans that good talent is coming through at Murray Park."
McKay has scored four goals in 30 appearances for the club this term and McCoist believes the teenager has more to offer.
"Barrie is still very young and he does things that young players do, but we have been very encouraged by him," he said.
"However, he is at a far greater stage in terms of his progression than we thought he would be which is great but, of course, he still has a long way to go."