Miami will receive a 2020 sixth-round pick that can become a fifth-round pick if conditions are met, sources told ESPN.
Arizona will need Drake's contributions in a hurry since David Johnson (ankle) and Chase Edmonds (hamstring) are dealing with injuries that will likely hold them out for at least Thursday night's game against San Francisco.
To fill the open roster spot, the Dolphins activated defensive tackle Robert Nkemdiche from the physically unable to perform list. Nkemdiche, who has recovered from his ACL injury, is eligible to play starting Monday night against the Pittsburgh Steelers.
Drake, the Dolphins' leading rusher, did not travel with the team for it's Monday night contest against the Steelers in anticipation of a trade. Multiple teams had called the Dolphins about Drake ahead of Tuesday's NFL trade deadline, sources told ESPN's Adam Schefter.
The speedy 25-year-old running back adds playmaking and dual-threat ability for a Cardinals team that's 3-4-1 and hoping to stay in the playoff picture.
For the Dolphins (0-6), the move is about continuing to stockpile 2020 draft picks for what is expected to be an aggressive offseason for general manager Chris Grier.
Miami now has 13 picks in the 2020 draft, including three first-round picks, two second-round picks and two expected compensatory selections. Their own selection, which could be used to select a franchise quarterback, is projected to be first overall.
Dolphins fans will remember Drake most for his playmaking ability, most notably his touchdown in the "Miami Miracle" last December in which he outran a stumbling Rob Gronkowski for the game-winner against the Patriots as time expired. That remains the Dolphins' most recent victory.
Drake, a 2016 third-round pick, is in his contract year, and there has been no progress on extension talks with the team this year. He was also recently supplanted by Mark Walton as the team's lead back. It seemed clear that Drake didn't have a long-term future in Miami, and both sides were prepared to move on.
Walton now should have the keys to the Dolphins' backfield. The 2018 fourth-round pick is putting together a solid sophomore year after a rough rookie year combined with an offseason of legal issues.
The Dolphins are likely to give 2018 fourth-round pick Kalen Ballage, undrafted rookie back Patrick Laird and rookie seventh-round pick Myles Gaskin bigger reserve roles with Drake gone from the rotation.
Drake has 348 total yards this season (174 rushing, 174 receiving) and also could provide kick-return capabilities for the Cardinals if they want to use him in that way. But his most important immediate role will be manning the backfield while Johnson and Edmonds deal with injuries.
The Dolphins have traded left tackle Laremy Tunsil, receiver Kenny Stills, defensive back Minkah Fitzpatrick, linebacker Kiko Alonso and Drake over the past two months as part of the team's extensive rebuild.
On Sunday, the Cardinals were also without running back D.J. Foster, who has missed the past two games with a hamstring injury. He was placed on injured reserve Monday.
Arizona signed running backs Zach Zenner and Alfred Morris on Wednesday to provide backup for Edmonds during practice. Zenner played just one offensive snap before Edmonds' injury Sunday, according to NFL Next Gen Stats. He finished with 19 offensive snaps. Morris, however, was a healthy scratch.
ESPN's Josh Weinfuss contributed to this report.