Mark Uth's 87th-minute strike, however, gave Hoffenheim a boost ahead of the return leg at Anfield next Wednesday, which Alexander-Arnold admits has left his side disappointed.
The full-back told BT Sport: "It's a thing of dreams to make your European debut for your boyhood club and especially to mark it with a goal.
"We're disappointed with the late goal but it's good to take a lead back to Anfield.
"I got egged on to take the free kick. I had the confidence to finally take it. We know it won't be an easy game but we have strategies to beat them."
Liverpool will be looking to seal a return to the Champions League group stages for the first time since 2014.