With Jurgen Klopp's men coming under pressure from the hosts in their Champions League play-off first leg, the 18-year-old fired a superb effort into the bottom corner to put his side in the driving seat.
Alexander-Arnold was delighted with both the goal and the win, although he conceded some disappointment at Mark Uth's late goal that keeps the tie alive, with James Milner's deflected effort having initially doubled the visitors' lead.
"It's a dream to make your European debut for your boyhood club and to make it with a goal as well was very special for me," he told BT Sport.
"But more importantly, we got the win. We'll still be a little bit disappointed with the late goal, but it's always good to take a lead back to Anfield.
Trent's first senior goal for the Reds...— Liverpool FC (@LFC) August 15, 2017
"I was egged on [to take the free-kick]. I had the confidence to take it and it paid off."
Jordan Henderson was full of praise for the young full-back, who also started in Nathaniel Clyne's absence in the Premier League on Saturday.
"He was brilliant last season," Henderson added. "He's come back and gone up another level again in pre-season.
"He's been brilliant. Last week [against Watford], I thought he was outstanding and then tonight, brilliant again.
"The free-kick is no coincidence. He has been doing that every day in training for the last few weeks – that’s why he's on them. He's done it on the big stage, so fair play to him.
"He's just got to keep working hard and putting in performances like that."