The league leaders opened the scoring after only four minutes when, after Clyde failed to clear a corner-kick, Russell McLean lashed the ball home.
McLean doubled Peterhead’s advantage midway through the second half with a penalty, conceded by Jordan Stewart, after which talismanic forward Rory McAllister added a slightly flattering third.
Second-placed Montrose kept pace with the leaders, overcoming Edinburgh City 3-0 at Links Park.
Lewis Milne fired the Gable Endies in front after four minutes, slotting home Martin Rennie’s through ball.
After Cammy Ballantyne was brought down, Milne doubled his and Montrose’s tally from the penalty spot in the 19th minute.
Edinburgh offered little in response before Montrose applied some gloss to the scoreline in the final minute courtesy of a sublime Rennie free-kick.
Montrose have a game in hand over Peterhead and will reclaim top spot if they take anything from Tuesday’s game at bottom side Cowdenbeath, who edged closer to confirming their place in the relegation play-off by only drawing 1-1 with Stenhousemuir.
Mark McGuigan headed home a Ruaridh Donaldson cross in the 38th minute to give the Warriors the lead, only for Cowdenbeath to level almost instantly thanks to Jordyn Sheerin’s close-range finish.
Elgin City climbed to fourth with an impressive 3-0 success against third-placed Stirling Albion at Borough Briggs.
Nathan Flanagan curled an outstanding opening goal for Elgin after just seven minutes.
The home side then proceeded to kill off the game with more than an hour to spare as Brian Cameron’s close-range strike in the 18th minute and Shane Sutherland’s ruthless finish six minutes later steered Elgin to a convincing and precious three points.
Seventh-placed Annan Athletic and ninth-placed Berwick Rangers played out a forgettable 0-0 draw at Galabank.
Berwick’s Steven Thomson went close with a curling shot early in the second period, while Annan’s Jack Brannan struck the bar in the 70th minute on a dull Dumfriesshire afternoon.