Liverpool flop in Florence
The in-form Montenegro forward ran unopposed on to Cristiano Zanetti's offside-splitting pass and coolly slotted the ball home to open the score in the 28th minute.
He doubled the advantage nine minutes later by directing Juan Vargas's diagonal thunderbolt into the net before his side rode out some intense pressure in the second half to claim a memorable victory.
The Italians, who lost their opener 1-0 at Olympique Lyon, have three points, as do Liverpool after they beat Hungarian champions Debrecen by the same margin two weeks ago. Lyon lead the group after two opening wins.
Five-times European champions Liverpool made a timid start, which encouraged the hosts to shake off their stage-fright and push forward. They were soon rewarded with two goals.
For a spell the visitors looked dazed by Fiorentina's speed and slick passing, but they were a very different proposition after the break.
Sebastien Frey saved shots from Yossi Benayoun and Dirk Kuyt and Lucas headed a corner just over the bar at the start of the second half, while Alessandro Gamberini produced an excellent challenge to deny Fernando Torres a shot in the 71st.
The Spaniard also came close with a volley and Steven Gerrard headed just over late on, but Fiorentina stood firm, showing no signs of being distracted by president Andrea Della Valle quitting last week.
Indeed, they seem to be thriving on the turmoil as they also moved to joint third in Serie A on Saturday alongside Inter Milan by beating Tuscan rivals Livorno.