The visitors started the game impressively but failed to turn their possession into chances. Their best fell to striker Chris Templeman after 22 minutes, but he could only send his back-post shot from a Chris Miller cross into the arms of keeper Paul Mathers.
Peterhead creted a string of chances in the first half, Allan Youngson going close on three occasions with Dougie Cameron seeing a shot saved and Mark Perry heading wide.
Morton almost went in front soon after the restart with John Adam shooting straight at Mathers from 25 yards with Peterhead responding with a 20-yard dipping shot from Youngson following a Barry Linn right-wing cross.
Peterhead finally made the breakthrough after 77 minutes. Martin Wood sent over a cross with the outside of his boot from the left wing and Scott Michie, only on as a substitute for less than a minute, shot high into the net from eight yards.
Morton had a great chance to equalise three minutes from the end when David Hagen brought down substitute Jason Walker, but Mathers dived to his right to save Peter Weatherson's spot-kick.