The French forward declined the option of a 12-month extension in Perth to sign a three-year contract with the Transylvanian outfit.
Tade, who scored six goals in 41 appearances for Saints last season, first moved to Scotland seven years ago when he signed for Forfar.
He also had spells with Stranraer, Clyde and Raith Rovers before moving to the Scottish Premier League with Inverness in 2011, and signed for St Johnstone after just 12 months in the Highlands.
The player was represented by Ricky Gillies during the negotiations and the former Scotland Under-21 international said: "This is a fantastic move for Gregory.
"He came to Scotland from France in 2006 and joined Forfar when they were in the Third Division. He worked his way to the SPL the hard way. Now he's got his move to a big club in Europe and it is thoroughly deserved.
"It's a fantastic story for a young guy from Nantes and shows what hard work can do.
"Cluj have been in the Champions League a couple of times in recent years and played Chelsea on one of those occasions. They have got big ambitions to get back there and that is something that has excited Gregory.
"And the move to Romania opens up a whole new market for him too. Russia and Ukraine are close by and have some really big clubs now.
"There's a chance he could catch their eye and earn himself another big switch."