Rubuta ci gaba ga wannan sakin layi - Kozma an zaɓi shi zuwa Triple-A don fara kakar 2012. An tuna da shi ga Cardinals a ranar 31 ga Agusta, 2012, lokacin da aka sanya Rafael Furcal a cikin jerin nakasassu. Kozma ya yi amfani da shi don ba Cardinals damar yanke hukunci a wasan na biyar na gasar zakarun kasa ta 2012 (NLDS) a kan Washington Nationals. A lokacin NLDS, ya buga wasan gida - na farko a cikin MLB postseason - kuma ya kori 'yan wasa biyar. Kozma ya buga.227 a lokacin NLCS
a kan San Francisco Giants kuma ya kara da wani sata tushe. Duk da samun nasarar wasanni 3 zuwa 1 a cikin jerin, Cardinals sun rasa jerin a cikin wasa na 7 da yanke hukunci.