The Wee Rangers had famously defeated the Glasgow giants in the Scottish Cup back in 1967 but a repeat shock never looked likely in this second-round clash.
It had taken Ally McCoist's side a while to find their attacking rhythm despite their dominance in possession, though Little tested Paul Grant when he raced onto a Lee Wallace pass and drew a save at the near post.
Little went even closer from a Lewis Macleod pass and watched his shot, though short on power, roll goalwards until Michael Dunlop hacked to safety at the last moment.
Sebastien Faure's impressive volley was then tipped over the crossbar but McKay finally made the breakthrough (60) less than a minute after entering the fray, the angled drive his first touch of the ball.
Little has supplied the pass for that one and killed the contest with a goal of his own two minutes later, meeting a Robbie Crawford cross with a header that gave Grant no chance.