The 25-year-old doubled his side's lead early in the first half as he kicked off his Anfield career with a goal, before strikes from Coutinho and Jordan Henderson sealed the win
Brendan Rodgers saw his side take an early lead inside 10 minutes after a goal from Daniel Sturridge, knocking the ball past Mitchell Langerak in goal after a smart flick from Philippe Coutinho.
And their lead was doubled just minutes later with a debut goal for Dejan Lovren as the Croatia international rose high from a corner to nod the ball into the net from close range.
The Reds resumed play after the interval and continued their dominance. Matthias Ginter, a new arrival at Dortmund, gifted a third after his loose pass was picked up by Raheem Sterling and dispatched by Coutinho.
But Liverpool then put themselves in an unassailable lead on the hour mark after Sturridge slipped the ball through for Jordan Henderson to tap home to make it 4-0.
The lacklustre visitors failed to heed an early warning of Coutinho slicing them open with a ball to Sterling and were punished when Sturridge broke the deadlock. Lovren fizzed a pass into Coutinho, who produced a wonderful flick around the corner to the striker, who displayed typical composure to put Liverpool ahead with an ice-cool finish.
Lovren ensured that it was a debut to remember shortly afterwards with his first goal for the club and Gerrard went close on the stroke of half-time as Langerak produced a fine stop to deny his top-corner-bound 25-yard free kick.
Rodgers's men picked up where they left off after the break, with Coutinho deservedly getting in on the act before Henderson rounded off the game's best passage of play with a simple finish on the hour.
Liverpool continued to threaten with Javier Manquillo - another defensive debutant - impressing at right-back, while Dortmund toiled.
Coutinho was withdrawn to a standing ovation with 20 minutes remaining of a game that suggested Liverpool are dealing rather better with the departure of their star striker than a Lewandowski-less Dortmund.