Jim McInally's men knew a victory would guarantee them the silverware and two first-half goals from Jack Leitch saw off the Spiders.
Closest challengers Clyde, who started the day two points behind, lost 2-1 to Cowdenbeath at Central Park and now face Edinburgh City in the play-offs.
Peterhead missed out on the last day of last season when Montrose took the title but the Balmoor side can now look forward to League 1 football in 2019/20.
