Liverpool's English midfielder Trent Alexander-Arnold celebrates scoring the 1-0 during the Champions League football qualifier match TSG 1899 Hoffenheim vs Liverpool FC in Sinsheim, Germany, on August 15, 2017.
PHOTO: AFP
However, Klopp insists Liverpool are far from certain of a Champions League group stage berth despite the victory.
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Goalkeeper Simon Mignolet saved a first-half penalty, whereas 18-year-old defender Trent Alexander-Arnold curled in a free-kick to give Liverpool the lead.
Hoffenheim pulled a goal back when replacement Mark Uth managed to beat the offside trap to score late on to give the Germans hope.
"I am happy with the performance, we'd have liked to have won 2-0 and the goal they scored just before the end isn't so favourable for us," said Klopp. "We could have scored one or two more goals and we need Anfield in the return leg. This thing isn't over yet."
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Liverpool's want-away attacking midfielder Philippe Coutinho sat out the play-off with a back injury.
Belgium goalkeeper Mignolet saved an early first-half penalty to deny hosts Hoffenheim an early lead in Sinsheim.
When Liverpool's Dejan Lovren brought down Germany winger Serge Gnabry, Hoffenheim's ex-Leicester City striker Andrej Kramaric stepped up for the spot kick.
The Hoffenheim forward squandered the golden chance, weakly firing his 11th minute spot-kick at Mignolet.
At the other end, Liverpool's Egypt striker Mohamed Salah fired just wide of the post on 15 minutes after sprinting clear and beating Hoffenheim goalkeeper Oliver Baumann.
Alexander-Arnold gave Liverpool the lead when his floated free-kick beat Baumann on 35 minutes.
However, Hoffenheim could have gone into the break level when striker Sandro Wagner hit the post after Mignolet blocked a Gnabry shot.
Liverpool should have made it 2-0 straight after the break when Baumann saved Roberto Firmin's shot from point-blank range after Sadio Mane's clever cross.
Substitute Milner put the matter beyond doubt when he snapped up Firmino's pass to the wing from a quickly taken free-kick.
He fired home off his left boot on 75 minutes as the shot clipped Nordtveit and went into the net.
"We deserved to at least of had a draw," said Hoffenheim coach Julian Nagelsmann. "We were brave, but we were a bit too open in the second half and risked too much. The way the game went, with the missed penalty, was unfortunate for us."
Elsewhere, Apoel Nicosia took a big step towards the group stages with a 2-0 win over Czech side Slavia Prague in Cyprus with Igor de Camargo and Efstathios Aloneftis scoring in the first ten minutes.
In Switzerland, Young Boys defender Kasim Adams Nuhu scored an own goal in added time to hand CSKA Moscow a 1-0 win.
Steaua Bucharest, reduced to ten men after Mihai Pintilii was sent off ten minutes from time, drew 0-0 at Sporting Lisbon.
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