The deal was announced after Hanigan agreed to a $10.75 million, three-year contract that runs through 2016 and includes a club option for 2017.
Tampa Bay sent minor league pitcher Justin Choate and a player to be named to the Diamondbacks. Arizona dealt left-hander David Holmberg to Cincinnati.
Rays executive vice president of baseball operations Andrew Friedman called a Hanigan "a tremendously talented defensive catcher" who can also help the Rays offensively.
"He takes a lot of pride in what he does behind the plate and we also like what he can do in the batter's box, especially against left-handed pitching," Friedman said. "He's a guy we've had our eye on for a while. and so when we had the opportunity to acquire him, we were aggressive to do so."
Hanigan, 33, is expected to become Tampa Bay's primary catcher, even though he and Friedman said they won't head into spring training with a preconceived notion. The trade was completed a day after free agent catcher Jose Molina was re-signed to a $4.5 million, two-year contract.
"We'll figure out matchups, we'll figure out what makes the most sense on any given night, but we feel like he makes our team better," Friedman said. "Whether that's 70 games, 80 games, 90 games, 100 games, I don't know yet. We'll figure that out as the season gets underway."
Bell will be a candidate to become the Rays' closer -- a job held the past two seasons by Fernando Rodney, who is a free agent.
For the Diamondbacks, giving up the hard-throwing right-hander was a cost-saving move. The deal saves the team $5.5 million of Bell's $9 million salary -- and Miami is paying the other $3.5 million as part of the three-team trade in October last year. Arizona general manager Kevin Towers gains flexibility in trying bolster depth and plug holes.
"We've still got some bench pieces we need to put together, We still have interest in bringing back Eric Chavez," Towers said, adding that the Diamondbacks also remain in the market for a starting pitcher and perhaps a corner outfielder with power.
He's also excited about the prospects headed for Arizona from Tampa Bay, particularly the player to be named.
"Someone we value a lot as a prospect," Towers said. "That's not to take anything away from Mr. Choate, but I would say that probably is the key player in the deal."
Hanigan is a .262 career hitter who spent the past seven seasons with the Reds. He batted .198 with two homers and 21 RBIs in 75 games in 2013, when he spent two stints on the disabled list -- one because of a strained left oblique muscle and the other a sprained left wrist.
Friedman is confident the catcher is healthy and "will return to the really good player he's always been."
Hanigan is looking forward to getting to know the pitching staff and play for manager Joe Maddon, who's led the Rays to the playoffs four of the past six seasons.
"The numbers speak for themselves in terms of the competitiveness of the team year after year. It's a great thing they have going," Hanigan said.
"Everybody I've talked to has great things to say about Joe and the organization, the atmosphere they create to compete," Hanigan added. "I'm just looking forward to helping the team out. They've got a lot of young arms. They've got a lot of good pitching. I'm excited to get to know these guys."
Bell, 36, has 168 saves in 10 seasons with the New York Mets, San Diego, Miami and the Diamondbacks. He led the majors with 132 saves for the Padres from 2009-11 and his 166 saves over the past five seasons are third the majors after Jonathan Papelbon's 173 and Mariano Rivera's 170.
Bell was 5-2 with a 4.11 ERA with 15 saves in 69 appearances for the Diamondbacks last season. He will have an opportuinity to win the closer's job in Tampa Bay, although Friedman stressed the Rays didn't necessarily acquire him with that in mind.
"I think I have a shot of winning that job, the ninth-inning job, but I see my role as being to come in to spirng training and show them what I can do," Bell said.
"They always say I don't run from run from anybody, I don't back down from anybody. I go right after everybody," the reliever added. "I kind of feel like I just want to help the team out the best way I can."
The Reds traded a catcher they no longer needed for a left-handed pitching prospect.
Devin Mesoraco, their first-round draft pick in 2007, emerged as an everyday catcher last season when Hanigan was hurt. Mesoraco played in a career-high 103 games, batting .238 with nine homers and 42 RBIs. Hanigan was on the disabled list twice and batted only .198 with two homers and 21 RBIs in 75 games.
Hanigan caught both of Homer Bailey's no-hitters in the last two seasons. The 33-year-old became expendable when the Reds decided after the season that they would make Mesoraco the starter. They signed catcher Brayan Pena to a two-year deal for $2,275,000, giving them a backup for Mesoraco.
"Devin Mesoraco will have the opportunity to become a front-line catcher for us," general manager Walt Jocketty said. "Holmberg provides us with the quality pitching depth that every team needs."
The 22-year-old lefty was a Southern League All-Star last season. He went 5-8 with a 2.75 ERA for Double-A Mobile, striking out 116 in 157 1-3 innings. He made his big league debut for Arizona against San Diego on Aug. 27.