The deal was worth $2.6 million, a source told ESPN.com's Pierre LeBrun.
Islanders general manager Garth Snow has been aggressive in pursuing goaltending help this offseason. Snow traded a fourth-round pick to Washington earlier this spring to acquire the rights for Jaroslav Halak and later locked up the netminder to a four-year $18 million contract in May.
Now, the Isles have additional help between the pipes with Johnson signing in New York.
The 28-year-old journeyman backed up Tuukka Rask in Boston last season with an impressive 17-4-3 record, 2.10 goals against average and .925 save percentage in 27 games played for the Bruins.
Pierre LeBrun of ESPN.com contributed to this report.