The Scotland international signed on the final day of the January transfer window in a £1million deal from Wolves and has already impressed the club captain in training.
Brown - who has likened Griffiths' left foot to "a wand"- reckons that the Scotland star has benefited from a few familiar faces in the Lennoxtown dressing room from international duty.
Brown said: "Leigh has been firing on all cylinders and looking really sharp in training. He is a quiet lad off the park but when he gets the ball to his left foot he comes alive.
"I was watching him in shooting practice and his left foot is a wand. As soon as he unleashes it, the ball seems to go into the back of the net.
"What he did at Hibs, with his 25 goals or so at Easter Road in a season, was phenomenal. Not a lot of people have done that at Hibs over the years.
"He takes the ball and tries to go one-on-one with people and so he has real belief and faith in himself and that is a good thing coming to a club like Celtic.
"You have to have belief in yourself at Celtic and Leigh has that. He has not had a lot of game time, but we believe Leigh will deliver for us."
With Brown driving Griffiths through from Edinburgh every morning and fellow-Scotland stars Charlie Mulgrew and Kris Commons already fixtures in Paradise, the Celtic captain has no doubt that all the new Bhoy needs now is match action.
Brown said: "Hopefully, it is a benefit to him that he has guys in the dressing room he knows from Scotland. Obviously there is myself, Charlie Mulgrew and Kris Commons, and he knows us all well enough.
"Certainly, when I joined Celtic it was a plus to have boys you know in the side and I think you can see that Leigh has fitted in well already.
"I'm his chauffeur every morning and he buys me a coffee, so he's sorted. All Leigh needs now is games."
When it came to this afternoon's highly-anticipated Scottish Cup fifth-round tie with the Dons, Brown admitted he is looking forward to catching up with a few familiar faces.
He said: "Aberdeen are an exciting team and they have players with real experience in the middle of the park with Barry Robson and Willo Flood sitting in there.
"Barry is maybe getting on a wee bit and he has adapted his game to cope with that. I have not played against Barry since he left here, but Willo will be in there snapping around my heels, no doubt.
"Then, further back they have Russell Anderson and clearly Niall McGinn was the last player to score against us domestically, so Aberdeen are a big threat.
"They will be on a high after reaching the League Cup final, but it comes to 90 minutes and who wants it more, and we must make sure that is us."
Adam Matthews is still not fit to return to the fray for Celtic, while there is a big doubt over James Forrest.