The Championship side dominated the first half but it was Montrose who almost broke the deadlock in the 20th minute when Lloyd Young lashed an effort against the crossbar.
The hosts eventually made their pressure pay - Kabba-Modou Cham firing home from the centre of the penalty area four minutes after the break.
Marc Fitzpatrick headed a second from close range in the 69th minute before the Ton made sure with a stoppage-time double.
Reece Hands and Aidan Fulton both converted from inside the penalty area - little more than a minute apart - as Morton marched into the third round of the competition for the first time since 2008/09.
- Sports & Recreation
- Marc Fitzpatrick