Bove, a Roma Academy rookie, scored the winning goal in the 63rd minute after following Tammy Abraham’s shot, which was saved by Leverkusen goalkeeper Lukas Hradecky.
The visitors nearly equalized at the end of a choppy second half, but Jeremy Frimpong’s 87th-minute shot was cleared from the line.
Roma, chasing a second straight continental title under manager Jose Mourinho after winning the Europa League last season, travel to Leverkusen next week for the second leg.
The winner of this match will face the winner of the Juventus-Sevilla match in the final.
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