Newsday also reported the Islanders have signed forward Jack Skille to a one-year, two-way contract.
TSN placed the total value of Johnson's contract at $2.6 million.
"Obviously to get a deal done with an organization like the Islanders, with a young group of players who are hungry to win, it's exciting for myself to join the team," Johnson told the team website.
Johnson, 28, went 17-4-3 with a 2.10 goals-against average and .925 save percentage in 27 games as the backup to Tuukka Rask with the Boston Bruins last season. He will fill a similar role with the Islanders, who previously acquired Jaroslav Halak from the Washington Capitals and signed him to a four-year contract.
"Hopefully for myself and Halak we can bring some stability in goal and keep moving forward with the group that we have," Johnson said. "It's exciting for myself to get to this organization and help the team win as many games as possible."
In 37 games during parts of four seasons with the New York Rangers, Phoenix Coyotes and Bruins, Johnson is 20-6-6 with a 2.07 GAA and .926 save percentage.
The Islanders also signed goaltender David Leggio to a one-year, two-way contract, according to ESPN.com. The 29-year-old had a 2.63 goals-against average in 43 games with the Hershey Bears last season and played one game for the United States at the 2014 IIHF World Championship.
The Islanders also signed forward Harry Zolnierczyk to a one-year, two-way contract, according to TSN; he'll make $600,000 if he's in the NHL, $300,000 if he's in the American Hockey League. He had two goals in 13 games with the Pittsburgh Penguins last season and 18 goals and 36 points in 57 games with the Penguins' AHL team.
Brennan, 25, won the Eddie Shore Award in 2013-14 as the best defenseman in the American Hockey League. In 76 games with the Toronto Marlies, the Toronto Maple Leafs' AHL affiliate, Brennan led AHL defenders with 25 goals and 72 points. He had 14 points in 14 AHL playoff games.
He last played in the NHL in 2012-13, with three goals and 10 points in 29 games with the Buffalo Sabres and Florida Panthers. In 40 NHL games with the Sabres and Panthers, he has four goals and 11 points.
Skille, 27, had four goals in 16 games for the Columbus Blue Jackets last season and 13 goals and 24 points in 22 games for the AHL's Springfield Falcone. He was the seventh player taken in the 2005 NHL Draft and has 29 goals and 53 points in 194 games with the Chicago Blackhawks, Panthers and Blue Jackets.
The Islanders also signed left wing Kael Mouillierat to a one-year, two-way contract. Financial terms were not released. The 26-year-old had 20 goals and 53 points last season with the St. John's Ice Caps, the Winnipeg Jets' American Hockey League affiliate.
He previously played for the Islanders organization during the 2011-12 season when he had 23 points in 44 games with the Bridgeport Sound Tigers, the Islanders' AHL team.