Morton into last eight
impose themselves on the game as the half progressed, Iain Russell flicking a header just wide of the post.
Ross, though, continued to be a thorn in the Sons side and he dragged a left-footed effort wide of the post after being released by Scott Tiffoney’s clever back-heel.
The Greenock side doubled their advantage on the restart through Australian defender Jack Iredale who was on hand to stab home the rebound after Gallacher had brilliantly parried Oliver’s shot.
Man-of-the-match Ross had a golden opportunity to secure Morton’s place in the quarterfinal draw but somehow headed over from five yards out with 20 minutes left.
Dumbarton pressed forward looking for a way back into the match but this left space for Morton to exploit on the break.
The almost did so when Gallacher brilliantly tipped substitute Bob Mchugh’s ferocious drive wide.
However, Mchugh sealed the victory when he sidefooted Tidser’s cross into the net.
Sons boss Steven Aitken admitted: “We lost poor goals and we can’t afford to give cheap goals away.
“It’s a difficult place to come against a side doing well in the play-off places.
“We have a semi-final next week and the league campaign is going okay so we just have to move on.”