The 30-year-old Wittgren was 2-9 with one save and a 5.05 ERA in a career-high 60 games for Cleveland last season.
In six seasons with Miami and Cleveland, he is 18-15 with five saves and a 3.75 ERA in 258 games. He has struck out 265 in 271 1⁄3 innings.
Wittgren has permitted only 19.8% of inherited runners to score since 2016, the sixth-best rate in the majors.
On Friday, right-hander Drew VerHagen and the Cardinals agreed to a two-year, $5.5 million contract.
The 31-year-old VerHagen went 10-10 over parts of six seasons in Detroit and spent the last two seasons in Japan.
The Cardinals are in line to begin the exhibition season on Friday vs. the Houston Astros in Sarasota, Florida.
The Associated Press contributed to this report.