The French forward had the option of a further 12 months in Perth but turned that down to pen a three-year deal with the Transylvanian outfit.
The 26-year-old will now link up with boss Mircea Rednic's squad at their pre-season training base in the Austrian Alps.
Tade - who finished last season with six goals from his 41 outings for the Saints - first moved to Scotland seven years ago when he signed for Forfar. He also had spells with Stranraer, Clyde and Raith Rovers before finally making it to the Scottish Premier League in 2011 with Inverness.
He moved on to St Johnstone after just 12 months in the Highlands but is now setting his sights on helping the 2012 Romanian champions seal a return to the Champions League.
The player was represented by Ricky Gillies and his ProLegal Group during the negotiations and the former Scotland Under-21 international told Press Association Sport: "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."
- Sports & Recreation
- Scottish Premier League