Cron had been designated for assignment by the Rays despite hitting 30 home runs and posting an .816 OPS last season.
Cron batted .253, slugged .493 and drove in 74 runs in his first season with Tampa Bay, but the Rays cut the 28-year-old last Tuesday before he entered his second season of arbitration eligibility.
He made $2.3 million last season while splitting time between first base and designated hitter.
The Twins have 39 players on their 40-man roster after claiming Cron.
The Associated Press contributed to this report.