The 29-year-old striker confirmed on his official website that he had signed a contract, which has an option for a renewal, after completing a routine medical.
Terms of the deal were not disclosed but local media have reported he would be earning in excess of the $250,000 per week he was receiving at UAE champions Al Ain.
A report on Gyan's website said the club would hold an official unveiling when Shanghai return from Sunday's league match at Guangzhou R&F.
Gyan was immensely popular during his four-year stint at Al Ain, where he scored 95 goals in 83 matches, but the Emirati team said the offer from the title-chasing Chinese side was too good to refuse.
Gyan's arrival marks the latest big-money signing in China, where the top clubs have splashed out millions of dollars on high-profile players such as Demba Ba and Paulinho, with Brazilian forward Robinho also expected to join their ranks.
(Reporting by Peter Rutherford in Seoul; Editing by Nick Mulvenney)
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