Former Gers forward Andrew Shinnie gave Terry Butcher's side the lead just before the half-hour mark in an evenly-matched first period.
Cally were the stronger team in the second half, and went further ahead through a Gary Warren header on 59th minutes.
The visitors' third came from the penalty spot after Gers substitute Anestis Argyriou brought down Shinnie in the box. The midfielder's younger brother Graeme stepped up and sent Neil Alexander the wrong way for a final score of 3-0.
Hearts also progressed to the semi-finals of the competition after defeating Dundee United 5-4 on penalties at Tannadice Park.