The 21-year-old yesterday completed his transfer from the Strathclyde Homes Stadium, putting pen to paper on a two-year deal.
And Ton boss Moore says he had no problem in making the Sons a small development payment for Wallace as he believes the versatile right-sided player has a great deal of potential.
He told the Tele: "Tony Wallace has signed his papers and that transfer has been registered with the league.
"We've actually signed him on a two-year deal because he's a young lad - he's only 21 - who has a lot of potential and who we believe can go further.
"He's not the finished article yet but we do think he's got something there and we've had to pay Dumbarton a wee bit of money to get him.
"And that's something I've got to say, Dumbarton were excellent about it. I spoke with their manager, Alan Adamson, and asked permission to bring Tony in for pre-season training.
"He made it clear that they would like to have him back but that they wouldn't stand in the boy's way, and the negotiations over the development fee were quick and smoothly conducted.
"But we're not putting any onus on Tony. He's young and is coming in to learn and to progress. It might take him three or four months to get into the swing of things.
"We believe the two-year deal will give him the chance to do that without putting any pressure on him."
Wallace particularly impressed the Morton management duo with his performances in the second half of last season, with Mark McNally personally watching him on several occasions.
And Moore admitted that it was assistant McNally's glowing reports that lead to him making the move to secure the player's signature.
He added: "Mark McNally has to take the credit for this signing. He went to watch Tony four or five times last season, and he shone for Dumbarton by all accounts, especially in the play-offs.
"We also did our homework and spoke to a lot of people. Jack Ross and Paul Nugent, who I had at Stirling for several years, were very complimentary about him.
"[Former Dumbarton manager] Jim Chapman actually signed him as a centre-half. He can play there, at right-back or in one in midfield, but we've primarily signed him for right-midfield.
"Tony's got a great attitude; he's hungry and willing to learn. He also has a great deal of faith in his own ability, and we were willing to put our faith in him too."