Ben Hall's ninth-minute opener was added to by Aidan Keena's first league goal for the club in the 68th minute as the hosts secured a first win in four games in all competitions.
Joshua Jack pulled one back with six minutes remaining but Clyde's first defeat in three left them a point above the relegation play-off spot.
Rory McAllister struck nine minutes into the second half and his seventh goal of the season was enough to earn three points.
Montrose let slip a two-goal lead and fell seven points off the pace after drawing 2-2 at Airdrie.
The hosts were 2-0 up at half-time after Cammy Ballantyne's 30th-minute opener and Sean Crighton's stoppage-time own goal.
Partick Thistle climbed above both teams into third with a 3-0 win at Peterhead.
East Fife are level on points with Thistle and Montrose after goals in each half from Ryan Wallace and Kevin Smith earned a 2-1 win at Forfar.
Jordan Allan had equalised early in the second half for Forfar, who had Barnard Coll sent off for a second booking late on and remain five points adrift at the bottom of the table.