General manager Jeff Luhnow announced the moves Friday, leaving the team with no arbitration-eligible players.
In his first season with Houston, Lowrie hit .244 with 16 home runs and 42 RBIs in 97 games. His .980 fielding was best among National League shortstops with at least 90 games played. He will make $2.4 million.
Norris, who will get $3 million, led the club in strikeouts last year with 165 in 168 2-3 innings. He was 7-13 with a 4.65 ERA.
Wright was 2-2 with a 3.27 ERA and a save in a career-high 77 appearances out of the bullpen. He'll get $1,025,000.