It took just three minutes for the home side to break the deadlock at Anfield through youngster Raheem Sterling.
The 17-year-old latched on to an incisive ball from Jose Enrique down the left before cutting inside to curl a shot into the bottom corner past Bernd Leno in the Leverkusen goal.
Liverpool’s second goal of the match came on the 29th minute from a free-kick won by Luis Suarez, who was brought down by Daniel Schwaab on the edge of the area.
The cleverly worked set-piece found Suarez unmarked in the area, who got the ball stuck under his feet before it was deflected into the path of Lucas Leiva, who side-footed home into an open goal from six yards out.
A goalkeeping blunder allowed Liverpool to get their third of the match on 65 minutes when substitute Carroll turned nicely on the ball before firing a routine shot from outside the area goalwards that Leno fumbled into the back of the net.
The visitors snatched a consolation through Gonzalo Castro on 75 minutes when he shrugged off Jamie Carragher and unleashed a fierce left-footed shot that sailed past Jose Reina in the Liverpool goal.