Duffy has made 130 appearances, and scored eight goals, in four years with the Premier League side but fell out of favour last season.
The centre-back, 28, bolsters Celtic's defence for their pursuit of a record 10th successive Scottish title.
"It's something I've always dreamed of and it's a special moment for me," Republic of Ireland cap Duffy said.
"It's all I knew growing up and I'm really excited. It's took a little while to get it done, but I'm over the moon."
Duffy, who is currently with the Ireland squad for their Nations League double-header against Bulgaria and Finland, underwent his Celtic medical in Dublin.
"He is leader and a warrior - someone with real ability and who will also give us that great physical presence," said. Lennon.
"I know too he is a player who will add great personality to the dressing room."