Cruz had missed four straight games due to the wrist strain and said the time of the year was a major factor.
The move makes room on the 40-man roster for right-handed reliever Austin Adams.
The Angels went 3 for 17 with runners in scoring position, leaving 14 men on base.
Norris turned in his longest outing since 2017.
The Twins (24-12) have the best record in the majors, backed by a surging rotation and a lineup that’s suddenly full of sluggers.
Corey Schneider made 32 saves for the Americans. They will face France on Sunday.