Marussia said in a statement at the Hungarian Grand Prix that the 22-year-old, who was previously reserve at rivals Caterham, had joined with immediate effect.
"The U.S. is an exciting market for the sport and Alexander has a great deal of support there, so it is nice to develop new links to that audience," said principal John Booth.
"Alexander will be an integral part of the team, immersing himself in our engineering activity so that he is fully prepared should he ever be required to deputize for one of the race drivers."
Although Marussia remain Russian registered, the cash-strapped team are based in Britain and independent from Russian sportscar maker Marussia Motors that had invested in it but closed its doors in April.
Their race drivers are Frenchman Jules Bianchi and Britain's Max Chilton.
Marussia started the season with the smallest budget in the sport but are currently ninth after Bianchi scored two points in Monaco. Caterham are 11th and last.
"Today is a good day and next step in my Formula One career," said Rossi, who hoped to drive in first free practise this season - most likely at his home race in Austin, Texas.
Rossi had been due to do that for Caterham but cut his ties with the team after Malaysian entrepreneur Tony Fernandes sold the ownership to unidentified Middle Eastern and Swiss investors.
(Reporting by Alan Baldwin, editing by Tony Goodson)
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