The South Korean, 23, bogeyed the 18th hole to finish on 17 under alongside 26-year-old Solheim Cup player Ciganda.
But Kim then birdied the par-four 18th in the play-off to seal her second LPGA title this season - after March's JTBC Founders Cup - and her fifth overall.
Wales' Becky Morgan finished tied 16th, with joint first-round leader Dame Laura Davies in a share of 43rd.
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Sei Young Kim defeated Spain's Carlota Ciganda in a play-off to win the Meijer LPGA Classic in Michigan.