All the goals came in the first half, with Lawrence Shankland scoring the winner after John Sutton had cancelled out Louis Appere's early opener.
It was a seventh straight league win for Robbie Neilson's men, whose next game is against Arbroath at Gayfield on Saturday.
The Red Lichties drew 1-1 with Ayr United at Somerset Park in Tuesday's other match - Michael McKenna equalised from the penalty spot after Stephen Kelly scored first for the Honest Men.
The Ladbrokes League 2 fixture between Albion Rovers and Cowdenbeath at the Reigart Stadium was postponed due to a waterlogged pitch.
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