The striker has been the League One side's top goalscorer this season, netting 17 times in all competitions, including 12 in the league.
The manager said in a statement: "The possibility of Leigh joining Celtic has been much publicised for some time now. With his family situation and the children he has up in Scotland, reluctantly we have agreed it is the right move for all concerned.
"We hope we can finish this season off strongly and then Leigh will have contributed to a successful season. It has been a fairly brief stay at Wolves but that sometimes happens with centre-forwards who move on. The deal is a good one for us and we can now move on with fresh players."
Griffiths signed a four-year deal with the Scottish champions earlier this evening.