Bove's calm finish settled an otherwise mediocre and sometimes ill-tempered affair, leaving the tie very much in the balance before next Thursday's return leg at the BayArena.
Attempting to get one over his former master Jose Mourinho, Xabi Alonso's side exploded out of the blocks at Stadio Olimpico and forced Rui Patricio into his first save with 42 seconds gone, as the Portuguese veteran comfortably kept out a first-time Robert Andrich strike.
The usual suspects in Moussa Diaby and Piero Hincapie both played their parts in setting up that chance with a pair of silky dribbles, and another one of Leverkusen's much-coveted talents in Florian Wirtz was a whisker away from breaking the deadlock in the seventh minute.
Having sauntered through a sea of Roma shirts on the edge of the box, Wirtz played a neat one-two with Adam Hlozek and attempted to side-foot home into the far corner, but the 20-year-old's strike trickled just wide of the post.
Following Leverkusen's rapid yet fruitless start, Roma began to get a foothold in the game with a steady period of possession, albeit with no real penetration through a staunch visiting backline.
A Lorenzo Pellegrini free kick very nearly provided the key to unlocking Leverkusen's defence in the 19th minute, as the captain's inswinging set-piece met the head of Roger Ibanez, but Lukas Hradecky reacted brilliantly to palm the defender's effort away before Bryan Cristante sent the rebound into the side netting from a tight angle.
As a closely-contested first half progressed, the game became very stop-start as players dropped to the turf all over the shop, one of whom - Odilon Kossounou - had to be withdrawn in the 35th minute after sustaining an apparent hamstring problem; Mitchel Bakker replaced his stricken teammate as Hincapie dropped into Leverkusen's back three.
The Germans' efforts to frustrate Mourinho's men continued to bear fruit in a fractious start to the second period, but Leverkusen's resistance was finally broken with 63 minutes played.
Strolling past Edmond Tapsoba as if the centre-back was not even there, Bove drove forward and laid off Tammy Abraham, whose shot on the turn was palmed back into the danger area by Hradecky, and Bove charged forward to steer the rebound into the bottom corner.
Leverkusen showed few signs of a revival against a well-organised backline, but a mix-up between Ibanez and Patricio presented them with a golden chance in the 87th minute - the former flapped at a ball into the box, elbowing his teammate in the process, and Jeremie Frimpong tried to capitalise on the chaos, but his goal-bound strike was blocked by the chest of a well-positioned Cristante.
Prior to next week's intriguing return leg, Roma will continue their quest to break into the Serie A top six away to Bologna on Sunday, while a trip to Stuttgart awaits Leverkusen in the Bundesliga.