The home side went in front after only 40 seconds, Scott Michie laid the ball back to Dougie Cameron and he shot into the net via the post from the edge of the penalty area.
As the snow began to fall the visitors pushed forward, but failed to pose a threat and were punished on the half hour.
Jamie Buchan sent over a corner kick from the left which broke to Bobby Linn who shot through a ruck of defenders on the goal-line from 12 yards out.
Peterhead went for more goals from the restart and Linn and Michie went close before Michie saw another shot held waist high by City keeper Kieron Renton.
The Blue Toon made the game safe after 65 minutes, Renton spilled the ball allowing Iain Good to cross from the left wing for Buchan to loop a header into the left side of the goal.
Four minutes from the end Elgin's Adam Nelson was red carded for violent conduct.