Green’s Dictionary of Slang

kick v.4

[abbr. kick the habit under habit n.]
(orig. US drugs)

1. [1920s+] (also kick the monkey) to stop taking an addictive drug.

2. [1960s+] to stop any form of addiction.

In phrases

kick off (v.) [1950s]

1. (US drugs) to sleep off the effects of drugs.

2. to withdraw from narcotic addiction.