Profar was likely to be non-tendered before the 8 p.m. ET deadline Monday, and Oakland wound up getting 25-year-old Allen, who has some pop.
Profar was acquired by the A's from the Rangers in December 2018. In his one season in Oakland, he batted just .218 but equaled his career best with 20 home runs and drove in 67 runs.
The 26-year-old was coming off a career year in Texas, where he hit .254 with 20 homers and 77 RBIs. He spent the first five years of his career with the Rangers. Current Padres general manager A.J. Preller was assistant GM with Texas during Profar's time there.
Allen, 25, played in 34 games during three 2019 stints with the Padres for his first major league experience. A fourth-round draft pick by the Padres in 2015, he batted .215 with four doubles and three RBIs while striking out 21 times to his six walks.
Allen also made his debut at the Triple-A level last season and has hit 20 or more home runs in each of his last three seasons in the minors.
In addition, infielder Greg Garcia agreed to a $1.5 million, one-year deal with the Padres that avoided arbitration.
Information from The Associated Press was used in this report.