Kazmir was traded to the Angels on Friday for left-hander Alex Torres and third baseman Matt Sweeney, and a player to be named later, who turned out to be Rodriguez.
Rodriguez has shown great power at Triple-A Salt Lake this season, batting .299 with 29 homers and 93 RBIs in 103 games. He ranked second in the Pacific Coast League in homers and slugging percentage (.616) and was tied for third in RBIs.
"I couldn't be going to a better team than Tampa Bay," said Rodriguez, who was in Tacoma and will fly to Durham on Wednesday. "With what they did last year, and they're making a playoff run again this year, I'm super, super excited."
"He's very exciting," Rays manager Joe Maddon said. "He's the kind of guy if you auger out one spot for him in the lineup and give him a bunch of at-bats, I think what he's done in the Minor Leagues he could do something similar in the Major Leagues."
Right-hander Joe Nelson was designated for assignment to make room on the 40-man roster for Rodriguez.
Rodriguez had to clear waivers before he could move to the Rays. If he had not, the deal would have been finalized after the season. Since he did clear waivers, he will first report to Triple-A Durham, but will be called up to the Rays in the next week so the team can get a good look at him.
"Good kid, good kid, man," Maddon said. "Really talented, can play a variety of different positions well. Probably if you had to pick one best position, it would be second base.
"He can play shortstop, corner outfielder, can also play center; throws well, runs well. Also has some good power. You look at him physically, you wouldn't see all that power coming out of him if he was standing here, but it's there."
Appearing in 71 games for the Angels the past two seasons, Rodriguez, 24, made an impression with his defense, his attitude and his booming bat.
Angels manager Mike Scioscia raved about Rodriguez after his performance against the Twins in the July 31-Aug. 2 series at the Metrodome. Playing left field in two games, he had two singles and a home run to center field, making several high-level plays with the glove.
"From the perspective of being an everyday player, he's opened up a new dimension being able to play the outfield every day," Scioscia said. "He's got enough thunder in his bat to play corner outfield. He can play second base every day, shortstop, third base. He's a baseball player, a terrific athlete.
"The way he handled himself on the defensive end [at the Metrodome] is a plus. We've seen glimpses of the power he has. He's putting it together."
A Miami native, Rodriguez batted .200 in 25 at-bats this season for the Angels with two homers and four RBIs. In 59 games last year, he batted .204 with three homers and 10 RBIs in 167 at-bats. Defensively, he excelled at second with Howard Kendrick and Erick Aybar both sidelined. The only knock against Rodriguez has been his high strikeout numbers. He struck out 62 times in 192 at-bats for the Angels.
A third-round pick in the 2003 First-Year Player Draft out of G. Holes Braddock High School in Miami, Rodriguez was primarily an outfielder but also played the infield. He said he even caught when he was a kid and volunteered to be an emergency receiver for the Angels.
Maddon knew Rodriguez from his days in the Angels' organization.
"He used to come to some Spring Training games on occasion, and I got to visit with him and liked him, obviously, a lot," Maddon said. "Our scouts really did like him. I think it just caps off a really nice deal for us."
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.