Socceroos back-up keeper Langerak, starting ahead of regular No.1 Roman Weidenfeller, was beaten by early strikes from Daniel Sturridge and new boy Dejan Lovren.
And further goals from the outstanding Philippe Coutinho and Jordan Henderson after the break gave the Premier League runner-up an easy victory.
Despite the scoreline, it was a memorable day for Amini as he came off the bench in the 77th minute for his first team debut as coach Jurgen Klopp showed his faith in the future of the former Central Coast Mariners whizkid.
But it is the performances of a new-look front three of Sturridge, Coutinho and Raheem Sterling that caught the eye as Liverpool boss Brendan Rodgers gleaned maximum satisfaction as it began life in the wake of Luis Suarez's big-money move to Barcelona.
Exhibiting pace, technique and relentless movement, the trio terrorised Dortmund throughout - and Rodgers' pleasure at their display is likely to be in sharp contrast to the feelings of the watching Ronald Koeman, who brings Southampton to Anfield for its Premier League curtain-raiser next Sunday.
It took just 10 minutes for Sturridge to open the scoring, as he latched onto a superb Coutinho flick before sliding the ball past Langerak with the outside of his left foot.
Lovren added a second just four minutes later, as he met an inswinging Steven Gerrard corner with a firm header from six yards.
Coutinho fittingly got on the scoresheet just three minutes after the interval, as he shot coolly beyond Langerak from close range after being fed by Sterling.
Liverpool saved the best for last, though, with a dazzling piece of interplay between the front three ending with Henderson tapping home.
With a DFL Supercup meeting against Bayern Munich - and former hero Robert Lewandowski - just three days away, Dortmund's abject performance will be of concern to Klopp.
The lacklustre visitor failed to heed an early warning of Phillipe Coutinho slicing them open with a ball to Raheem Sterling, and were punished when Sturridge broke the deadlock.
Lovren fizzed a pass into Coutinho, who produced a wonderful flick around the corner to Sturridge and the striker displayed typical composure to put Liverpool ahead with an ice-cool finish.
Lovren ensured it was a debut to remember shortly after 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' 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.