The 21-year-old, dubbed by some as the "new Robert Lewandowski" has netted seven goals in 18 appearances for Polish top-flight side Jagiellonia Bialystok this season.
He is Celtic's first signing of the January transfer window, and arrives for a fee of around £3.5 million on a four-and-a-half year deal.
Klimala has been capped five times by Poland at Under-21 level, with two goals.
Celtic announced his arrival by posting a video featuring club mascot Hoopy the Huddle Hound on the video app TikTok.