Despite dominating for long spells at Cappielow, the Ton looked to be heading for a frustrating result as they struggled to break through.
They now remain four points behind Irn-Bru First Division leaders Partick Thistle after substitute Lewis Hawke grabbed his first goal for the club in the dying seconds.
Moore said: "It was a difficult game, not too unlike the Hamilton game here the other week. We had that much of the ball and we were thinking, 'Oh no, here we go again'. But we kept plugging away and wee Lewis has scored.
"It was a great finish and Lewis is showing great signs of maturity and now gives us options."
Hawke, the son of former Ton hero Warren, was given his chance to shine after Kyle Wilkie was forced off through injury just minutes after replacing Willie Dyer.
It was to prove a stroke of luck for Morton as he slid in to force the ball over the line with two minutes remaining and finally broke the stern Cowdenbeath resistance.