But Allan Moore's men came out firing on all cylinders after the break with Tony Wallace breaking the deadlock before David O'Brien added the clincher.
Summer signing Wallace admitted: "We started the second half at pace and that was down to a half-time rollicking from the gaffer.
"We were much better after the break and it was great for me to score a goal in front of the Cappielow fans who were great.
"I am excited about playing full-time football and I think the First Division is going to be difficult to predict with a lot of good teams."
Boss Moore admitted that Morton had to increase the pace against the Coatbridge squad who clearly came to stifle his side.
The Cappielow gaffer said: "We were a bit one paced in the first half and I felt we needed to get our full backs into the play.
"I told them to try to get two goals and then we could enjoy the second half a lot better and it turned out well.
"It was that 15 minute spell after half-time where we played our best football and won the cup tie.
"In the pre-season games against Premier League opposition we knocked the ball about well but Rovers never let us do that.
"They were in our faces and shutting us down and although I never felt we would lose a goal we had to do more in the wide areas."