Morton, who sit second bottom in Championship One having won just once in six games in the league, beat the Scottish champions thanks to a 97th-minute penalty from Dougie Imrie.
And Moore told Sky Sports News: "It's been a great feeling this morning going out and picking up the papers.
"I'm very proud considering our league form has been so indifferent this season. To a man all the boys turned up and they raised their game."
Morton, who have not beaten the Hoops at Parkhead since 1971, were forced to survive a late bombardment as the home side laid siege to the visitors' goal.
"The last five minutes was very difficult. Celtic put a lot of balls into our box but we stood up well and our keeper was excellent on the night," added Moore.
"I was really proud of the way the players stood up to a barrage of long balls into the area.
"Celtic had 27 corners that we had to deal with but we dealt with them comfortably.
"The dressing room after the game was incredible and I have to also thank the fans.
"It's been a long time since we had a result against Celtic."