The 34-year-old Nixon, known for his gritty playing style, hit .309 with 10 homers and 31 RBIs in 58 games with Triple-A Tucson after failing to make the Diamondbacks' big league roster out of spring training.
Arizona gets cash or a player to be named in the deal.
Nixon is expected to fill the void left by Moises Alou, who went back on the disabled list Friday with a lingering calf strain.
"With Alou down, it's a left-handed bat," said Mets general manager Omar Minaya, who expects Nixon to join the team for Saturday's game against Texas. "I checked around with a lot of guys who played with him and they all speak highly of him."
One of them is Mets starter Pedro Martinez, who was teammates with Nixon from 1998-04 and together helped the Red Sox win the 2004 World Series.
"He looks like a grouch all the time," Martinez said, suppressing a laugh. "But he's a good guy. You'll see."
Nixon hit .251 with three homers and 31 RBIs in 99 games last season with Cleveland, but was at his best in the playoffs. He was 5-for-11 with a homer and three RBIs against the New York Yankees and the Red Sox.
Nixon had an option in his contract that would have made him a free agent June 2, a deadline that he and Arizona extended earlier this month.
"In the business of signing players who have had such illustrious careers, staying to be a protection guy had to be a collaborative effort," Diamondbacks general manager Josh Byrnes said. "He's not going to spend the whole year in Triple-A when there was opportunity elsewhere."
Nixon has a .275 batting average in 11 big league seasons, 10 with the Red Sox. He had his best year in 2003, when he batted .306 with a career-high 28 homers and 87 RBIs.
"Everything I've heard about the guy, the guy is a gamer," Minaya said. "I've had this in the back of my mind for a while."