Midfielder Michael Tidser got the only goal of game to see off Raith Rovers at Cappielow, collecting Martin Hardie's knock down and scoring to ensure there would be no slip-up. It was a rare highlight as the home side struggled to find their rhythm.
But it proved a crucial intervention as Morton saw promotion rivals Partick Thistle edge out Cowdenbeath while Dunfermline crashed at home against Livingston.
The Jags play one of their two games in hand over Morton at the weekend when they face the Blue Brazil once again.
And Ton boss Moore was pleased with the efforts of his team.
He said: "Raith had a few clear-cut chances and fortunately they never took them and their keeper Derek Gaston was in fine form.
"You need a bit of luck at this stage of the season, especially when you don't play well.
"It was another win we had to grind out, but getting the three points in the most important thing and, thankfully, we were able to take our opportunity when it came.
"That is a big factor when you are trying to pick up points and challenge for titles.
"Unfortunately, Kevin Rutkiewicz hurt a knee while he was clearing the ball off the line and had to come off.
"But we hope he will be okay. It is important to have him available because he has been terrific this season.
"Thistle will have a hard game this weekend while we are playing Dundee in the Scottish Cup.
"It is difficult to play the same team back-to-back like they are with Cowdenbeath and the surface isn't the best.
"So we are hoping they can do us a wee favour and take points off them. We will wait and see how they get on."