Van Persie's signing and introduction ceremony took place at the club's Sukru Saracoglu stadium in Istanbul attended by over 10,000 jubilant fans.
The Dutch international is believed to have put pen to paper on a three-year deal worth 4.7 million euros (£3.4m, $5.2m) although financial details have yet to be disclosed.
"I am now part of the Fenerbahce family," van Persie told fans who created a deafening atmosphere and lit flares around the stadium. "I am happy to see so many great fans".
"I promise I will do everything in my power to make Fenerbahce win as many trophies as possible to make these fans happy."
His signing draws the curtain on three up-and-down years at Manchester United which began in a blaze of promise but ended in injury problems and faltering form.
Van Persie, 31, made 105 appearances and scored 58 goals at Manchester United after joining in August 2012 following his eight-year stint at Arsenal.
But his last season was marred by an ankle ligament injury and speculation of tension between himself and Manchester United boss Louis van Gaal, who had guided van Persie and his countrymen to third place in the World Cup.
Fenerbahce hav given their new flagship signing -- one of the best known players to arrive in Turkey in years -- the red carpet treatment and he was given a hero's welcome by hundreds of fans at the airport on Sunday.
Fenerbahce -- runners up to hated Istanbul rivals Galatasaray last season in the championship -- have gone on a spending spree in a bid to dominate the Super Lig and be a real contender in Europe.
Last week they signed Portuguese star Nani also from United for six million euros (£4.26m, $6.7m) and in June bought Danish centre-back Simon Kjaer from French side Lille.
Fenerbahce's flurry of transfer market activity is being masterminded by its well-respected new Italian sporting director Giuliano Terraneo who arrived in May on a mission to make the club a European football force.
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