Rookie full-back Alexander-Arnold demonstrated a cutting edge lacking in some of his more experienced team-mates with an expertly taken set-piece that found the bottom corner of the home net in the 35th minute to give him his first goal for the club.
Milner took advantage of acres of space on the left to find the top corner via a touch from a home defender just 10 minutes after coming on as a second-half substitute.
But defensive frailties again hampered the Reds as Mark Uth was found by a long ball into the box and fired in largely unchallenged in the closing stages.
It could have been worse for Jurgen Klopp’s side. In just the 11th minute, Simon Mignolet had prevented European debutants Hoffenheim taking the lead with a neat save to push away Andrej Kramaric’s poor penalty after Dejan Lovren had upended Serge Gnabry in the box.
Liverpool will look to seal a return to the Champions League group stages for the first time since 2014 at Anfield next Wednesday.
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