The deal is for two years at $11 million and worth up to $14.5 million, including $11 million guaranteed, a source told ESPN's Adam Schefter.
The Raiders opted to make a significant investment in Drake two years after drafting Josh Jacobs in the first round in hopes of forming a strong duo at the position.
Drake, who turned 27 in January, returned to the Cardinals in 2020 on a one-year, $8.4 million transition tender after getting traded to the team by the Miami Dolphins midway through the 2019 season.
In his first full year with the Cardinals, Drake ran for 955 yards and 10 touchdowns, while catching 25 passes for 137 yards. He missed one game, against Miami, which likely prevented him from reaching 1,000 rushing yards for the first time in his career.
But last season was his third straight with 1,000 all-purpose yards, proving he's among the most versatile running backs in the league.
A third-round selection by the Dolphins in the 2016 draft, Drake has rushed for 3,130 yards and 27 touchdowns and has 169 catches for 1,244 yards and 6 touchdowns in his career.
ESPN's Josh Weinfuss and The Associated Press contributed to this report.