The left-handed reliever has spent the last four seasons with the New York Yankees. He was 5-2 with a 3.23 ERA in 2013.
Logan will make $4.75 million next year, $5.5 million in '15 and then $6.25 million in '16.
The 29-year-old Logan is a big addition for a Colorado bullpen that had an NL-worst 4.23 ERA last season. The team also added reliever LaTroy Hawkins, who is expected to compete for the closer's job.
To make room for Logan on the 40-man roster, the team designed right-hander Collin McHugh for assignment. McHugh appeared in seven games last season with the Rockies and New York Mets, going 0-4 with a 10.04 ERA.