Woodward received a three-year contract with a club option for 2022. He is to be introduced at a news conference Monday.
Woodward, 42, will take on his first managerial job after three years with the Dodgers, who won the National League pennant the past two seasons.
After retiring in 2012, he joined the Mariners' organization and served as infield coach from 2014 to 2015 before signing on with the Dodgers.
The Rangers are coming off a dismal 67-95 season that saw them finish at the bottom of the American League West.
Woodward will replace Jeff Banister, whom Texas fired in late September as the team stumbled to its first consecutive losing records in 10 years, after winning division titles in each of Banister's first two seasons.
The hiring ends a process during which general manager Jon Daniels and the Rangers interviewed more than 10 candidates, including bench coach Don Wakamatsu after he served as interim manager the final 10 games of the season.
Woodward was teammates with Adrian Beltre for a short period during the 2009 season with the Mariners. Beltre, who has 3,166 career hits, just completed his 21st big league season and hasn't said whether he will return to play for the Rangers again next year.
The Associated Press contributed to this report.