The 41-year-old reliever and the Jays agreed to the deal on Dec. 30. The contract includes a $3 million option for 2013.
Infielder Mark Teahen was designated for assignment to make room on the 40-man roster.
Oliver, who is a modified Type B free agent, will net the Rangers a supplemental first-round Draft pick as compensation, but the Blue Jays will not be forced to surrender one of their own selections.
Toronto had been searching all offseason for a proven left-handed reliever to use in late-inning situations. The Blue Jays only had lefty Luis Perez under contract for 2012, but he is more suited to pitch in the middle innings and long relief.
Oliver will be used in a setup role with right-hander Casey Janssen to help bridge the gap from the starting rotation to recently acquired closer Sergio Santos. Anthopoulos also is expected to continue to explore ways to add another veteran right-hander into the mix.
Oliver was considered the top left-handed reliever on the market. He appeared in 61 games for the Rangers and was 5-5 with a 2.29 ERA that was 11th best among American League southpaw relievers with 50 innings pitched.
In 51 innings he struck out 44, allowed 47 hits and walked 11. He has been in the playoffs six straight seasons with the Mets, Angels and Rangers.