The tie was goalless from the first leg in midweek and it was Peterhead who took the lead with what turned out to be the winning goal in the 24th minute.
Bobby Linn sent Martin Wood clear and he was brought down by Morton defender Chris Millar, with Dougie Cameron scoring from the penalty spot sending keeper David McGurn the wrong way.
Morton only had themselves to blame for not at least taking the game into extra-time. They dominated the match virtually from start to finish but failed to create enough clear-cut chances and squandered those that they did.
The game was played at cup-tie pace in front of a record crowd at Balmoor Stadium.
There were few chances created until the 21st minute when Morton's James McAlistair saw his shot deflected out for a corner kick. A couple of minutes later Derek Lilley turned and shot into the arms of home keeper Paul Mathers.
Early in the second half Ryan Harding went close for Morton, thundering a free-kick straight at Mathers, but the Blue Toon could have wrapped up their victory 15 minutes from the end.
Substitute Martin Bavidge curled a shot from the left-hand side of the penalty area over the crossbar.