Steven Doris showed his predatory instincts to fire Stirling ahead after 16 minutes as he was the quickest to react in a goalmouth scramble to slam the ball home.
Sean Dickson made it 2-0 six minutes later with a powerful header from Mark Lamont's corner as the hosts took control of proceedings.
Doris all but sealed the victory four minutes into the second half as he collected Kevin McKinlay's through-ball before firing past Ross Doohan.
Craig Beattie made it 4-0 in the 73rd minute before Kyle Lander tapped home from close range with seven minutes remaining.
Beattie netted his second with virtually the last kick of the game as Stirling reached the fourth round, where they will host Inverness Caledonian Thistle.