Roma youth product Bove stabbed in the winner on the rebound in the 63rd minute at Olimpico Stadium for his first European goal.
Roma, bidding to win back-to-back European titles under coach Jose Mourinho following last season's Europa Conference League triumph, travel to Leverkusen next week for the return leg.
The winners will face Juventus or Sevilla, who drew 1-1 in their first leg on Thursday, in the final in Budapest on May 31.
It all started according to plan for Xabi Alonso's Leverkusen, who have not made a European final since their 2002 appearance in the Champions League showpiece.
In a lively start they came close twice, with Robert Andrich's shot saved after 40 seconds and Florian Wirtz wasting a golden chance when he fired wide.
The hosts, missing several players through injury and with Paulo Dybala and Georginio Wijnaldum not fully fit yet and on the bench, responded with Roger Ibanez' powerful header which Hradecky palmed wide with a superb reflex save.
They broke the deadlock just past the hour when Bove started a move and stabbed in on the rebound after Hradecky had saved Tammy Abraham's shot.
The visitors almost bagged an unexpected equaliser towards the end of a scrappy second half, but Jeremie Frimpong's 87th-minute shot was blocked on the line.
AAP
