The former Hibernian striker travelled north this morning after Neil Lennon's third bid for the Scotland international was accepted by the League One club.
Griffiths agreed terms on a four-year contract and put pen to paper on the deal after passing his medical in Glasgow.
"I'm over the moon," Griffiths told the Celtic website. "It's been a long time coming. Ever since I heard Celtic were interested there was only one place I wanted to play and I'm just glad it's done now.
"It was the middle of December and my agent gave me a phone and said Celtic could be interested in January and I have just been counting down the days.
"Unfortunately, it's taken to the last day but I'm just glad that it's been done now and I can't wait to get started.
"We are the Scottish champions to begin with, and this is a massive club with a massive fanbase and a massive stadium. I have watched the Champions League nights on the TV and there is no better place to play. This is where I want to be.
"I was just really keen. I know what it takes to become a Celtic player. I had a great season last year at Hibs and then I have went down back the road and started scoring goals as well.
"I want to bring goals - that's what I am all about and what I love doing. I did it last season for Hibs and have done it for Wolves and now it's my time to shine in the Hoops and hopefully I can do it.
"I am just thankful to get my chance here and hopefully I can prove that I am a good signing and start scoring goals for fun."
Lennon is aware that the reaction from some Celtic supporters to the proposed signing was not all positive, but he said: "I think it has been somewhat over the top.
"He is a young man. He has had a few off the field issues, as we all do. Sometimes they get blown out of proportion.
"So I will manage him as best I can and he has to concentrate on his football and if he does I am sure he will be a very good acquisition for the club.
"I have always liked his attitude on the field and having spoken to a lot of people who have played with him and worked with him, managers, they have spoken very highly of him.
"He is an engaging personality, he is quite quiet. Like most people who come with reputations he is not what you would expect.
"He seems quite a focused kid, he knows what he wants and he is delighted to be here. So we are very pleased.
"We think he is value for money. He's a prolific scorer, knows the environment here and I think we can make him better."