The Reds, again without wantaway star Philippe Coutinho, dug in against tough opponents and found inspiration in the unlikely form of their teenage right-back, before James Milner's effort - deflected in off Havard Nordtveit - widened the margin and Mark Uth hit back.
For 35 minutes, European debutants Hoffenheim delivered the type of high-energy display so readily associated with Klopp, who could hardly have complained had Andrej Kramaric's penalty handed the hosts the lead and not invited a simple Simon Mignolet save.
But Alexander-Arnold, with a confidence belying his tender years, swept an exquisite free-kick into the corner of the Hoffenheim goal 10 minutes before half-time and Julian Nagelsmann's side never really recovered.
Liverpool squandered numerous further chances and looked set for a nervy finish until Milner's cross-cum-shot drifted over Oliver Baumann via Nordtveit to seemingly secure victory at a stadium where Hoffenheim were unbeaten in the Bundesliga last season.
However, substitute Uth drifted in behind the visitors' defence with three minutes remaining to lash a superb effort past Mignolet and keep the German side in the tie going into the second leg next week.
Liverpool's back line, as in an entertaining 3-3 draw at Watford on Saturday, looked shaky from the off and Dejan Lovren had already survived one penalty appeal for handball when he clumsily clipped Serge Gnabry in the area and referee Bjorn Kuipers pointed to the spot.
But Kramaric's tame penalty bailed out the Reds defender, allowing Mignolet to firmly palm clear, and Mohamed Salah should have then put the visitors in front as he ran from the halfway line and prodded a tired finish past the right-hand post.
35% - Simon Mignolet has saved 7 of the 20 penalties he's faced in all competitions since joining Liverpool (excluding shootouts). Safe.— OptaJoe (@OptaJoe) August 15, 2017
The frantic early pace soon slowed but Hoffenheim's relentless pressing did not, with Jordan Henderson caught out and Kerem Demirbay crossing for Pavel Kaderabek to lash wildly over the top.
However, a moment of outstanding quality from Alexander-Arnold halted the hosts in their tracks, the 18-year-old firing a magnificent 25-yard free-kick beyond Baumann.
The response was strong and Mignolet blocked well from Gnabry before Sandro Wagner drilled the rebound against the outside of the post.
But back came Liverpool and Lovren nodded just wide on the stroke of half-time, with Roberto Firmino, against his former club, and Sadio Mane each then testing Baumann after the restart.
Predictably, with the clock winding down, Hoffenheim's efforts to preserve their fine home run became increasingly desperate and further chances instead fell the visitors' way as Baumann saved first from Salah's shot and then from Lovren's header.
With 16 minutes remaining, though, the second goal did arrive; Milner, in space on the left, appeared to look for a team-mate with his right-footed cross but instead found Nordtveit's chest and subsequently the far top corner.
A fine late Mignolet block kept Demirbay at bay, but Uth then thrashed past the goalkeeper to halve the deficit and keep Hoffenheim in the hunt in the closing stages of this home leg.
And the hosts should have equalised in stoppage time when Benjamin Hubner nodded Demirbay's set-piece inches over the crossbar, but Liverpool held on.
Key Opta stats:
- Liverpool remain unbeaten in their nine away Champions League qualifying matches, with seven wins and two draws.
- Simon Mignolet's early penalty save was his fourth from his last seven penalties faced in all competitions – in total he's saved 35 per cent of the penalties he’s faced since joining Liverpool (7/20).
- Trent Alexander-Arnold’s goal was his first for Liverpool, in his ninth start in all competitions.
- This was Hoffenheim’s first home defeat in any competition since the closing day of the 2015-16 Bundesliga season (1-4 vs Schalke).
- Liverpool scored more than once for the first time in their last nine European away games, since a 2-2 draw against Ludogorets Razgrad in the 2014-15 Champions League.
- Mark Uth’s strike was Hoffenheim’s first ever goal in European competition.