The deal is worth $1.5 million and includes incentives that could bring its total value to a maximum of $3.5 million, a league source told ESPN's Adam Schefter.
Foster's agent, Mike McCartney, announced the news Monday afternoon when he tweeted congratulations to the former Houston Texans star.
Foster, a four-time Pro Bowl running back, discussed why he joined the Dolphins later in the day on a conference call.
"They're a young team, a hungry team," Foster said after the Dolphins officially announced the signing. "[They have] a lot of talent on both sides of the ball. I think they have the right head coach and the right people in the front office. They're all committed to winning there."
Foster will join a Dolphins backfield that is in flux after Lamar Miller, Miami's leading rusher last season, signed with the Texans. Jay Ajayi and rookie Kenyan Drake currently are atop the Dolphins' depth chart at running back.
The Texans did not offer Foster a pay cut before releasing him on March 3. Foster, who has a history with injuries, turns 30 in August.
The Texans were set to spend nearly $9 million in cap space on Foster's salary, but Texans general manager Rick Smith said the decision to release Foster would not be based on his salary.
"The time that I spent in Houston was an amazing time," Foster said. "I'm grateful for the people that I met, my family that I made there and my family that lives there now. I got nothing but love for the people and fans of Houston. This is a new chapter of my life."
Foster played in only four games last season, hampered by two significant injuries. He suffered a torn groin in training camp that caused him to miss three games. He returned for four games, but then suffered a torn right Achilles tendon in the final five minutes against the Dolphins on Oct. 25.
Since becoming the Texans' starter in 2010, Foster has had a torn meniscus, multiple hamstring injuries, a knee injury, a back injury that required surgery, a minor groin injury, a major groin injury and a torn Achilles tendon. In that time, Foster has carried the ball an average of 20 times per game, the highest average in the NFL.
Foster went to the Texans as an undrafted free agent in 2009. He spent most of that season on the practice squad, being elevated to the active roster in November.
The 2010 season marked his breakout year. Foster rushed 33 times for 231 yards in a season-opening win over the Indianapolis Colts, sealing his role as the Texans' starting running back. He won the rushing title that year, gaining 1,616 yards on 327 carries, and went to his first of four Pro Bowls after that season.
ESPN's Adam Schefter, Tania Ganguli, James Walker and Mike Rodak contributed to this report.