Graham has penned a one-year contract after leaving Dunfermline Athletic following their relegation from the Clydesdale Bank Premier League.
The 28-year-old - who has also enjoyed spells at Stranraer, Gretna and Hamilton Accies - was a regular last season on his return to the top flight.
And Ton boss Moore admits he has been a fan of the wide man ever since a blistering stint with Stranraer where he scored 33 goals in 52 league games.
Moore told the Tele: "I'm really pleased to bring David on board. I have liked the look of him ever since he was with Stranraer.
"He is a good experienced player and is another piece of the jigsaw for us.
"The squad is coming together nicely.
"David is a really hard worker too - he didn't score too often last season but he puts in a lot of effort in wide areas.
"He also gives us options in attack as he can play on the right side, on the left and he is also comfortable playing up front.
"So that gives us good options and that is important to have. Hopefully he can come in and hit the ground running with us.
"David will cause other teams problems - he has a trick in his bag and is no slouch. I am pleased he has signed with us."
Right midfield proved to be a problematic area for Ton last season - with the likes of Paul Di Giacomo and Iain Flannigan struggling for fitness and form.
But Moore hopes the arrival of Graham will offer real threat from wide areas, with fleet-footed David O'Brien operating on the left-hand side.
He added: "We have O'Bri on the left and I think the arrival of David Graham gives us a good threat from both flanks.
"That was maybe a bit of a problem for us last season and we didn't really create a lot for our strikers to feed off.
"We have Peter MacDonald signed on up front again with wee Archie Campbell and Peter Weatherson so hopefully they can benefit."