Science prizes purportedly reward innovation and explorations of new phenomena. Yet, in practice prizes may inadvertently divert resources from similarly impactful but less celebrated scholars. Despite this paradox, knowledge of how prizewinning relates to innovation is nascent even as prizes proliferate widely. Analyzing 2,460 worldwide prizes, we compared the innovativeness of over 23,000 prizewinners and matched non-prizewinners whose performance records were statistically equivalent up to the prize year. First, we find that prizewinners are more innovative. Their research is more likely to combine existing ideas in new ways, integrate a topic's historical and contemporary thinking, and incorporate interdisciplinary perspectives. Second, although prizewinners and matched non-prizewinners have statistically equivalent impact and productivity records up to the prize year, at about five years before the prize, prizewinners' papers become more innovative than their matched peers, a difference that widens each year, peaks during the prize year, and then persists for the remainder of their careers. Third, network embeddedness predicts unusual innovativeness. Compared to non-prizewinners, prizewinners' collaborations are shorter in duration, encompass wider exposure to unfamiliar topics, and involve coauthors whose networks minimally overlap with each other. The implications of the findings for the efficacy of reward systems and innovation in science are discussed.