Teenager Trent Alexander-Arnold curled in a magnificent free kick on his European debut to set Liverpool on the way to a 2-1 win at Hoffenheim which put them on the brink of Champions League group stage place on Tuesday.
Liverpool goalkeeper Simon Mignolet saved an early penalty and James Milner`s shot was deflected into his own net by Havard Nordtveit for the English Premier League side`s second goal in an end-to-end match.
An astonishing last-minute own goal by Young Boys defender Kasim Nuhu handed CSKA Moscow a 1-0 away win while Sporting were held to a goalless draw at home by FCSB, formerly Steaua Bucharest, in two of the other Champions League playoff ties.
Alexander-Arnold, 18, who only made his Premier League debut in December, curled a free kick through a gap in the Hoffenheim wall 10 minutes before halftime as Liverpool survived a rocky start.
It could have been very different if Andrej Kramaric had converted a penalty for the Bundesliga side in the 12th minute but he saw a weak effort saved by Mignolet.
Liverpool had another let-off just before halftime when Mignolet saved from Kramaric and Wagner rolled the rebound wide of the post.
Hoffenheim, attempting to qualify for the group stage for the first time, had more possession but Liverpool were more threatening and went further ahead in the 77th minute.
The hosts lost possession trying to play their way out of defence, the ball was switched to substitute James Milner on the left and his shot from a narrow angle deflected off Nordtveit`s chest and into the top corner.
Mark Uth pulled one back with three minutes left, chesting the ball down and drilling a low shot into the corner of the net, and Hoffenheim should have equalised in stoppage time but Benjamin Huebner sent a free header over the bar.
"It was a little dangerous in the first half, but as always when we play football, we found spaces in between and created wonderful chances," said Livepool manager Juergen Klopp.
"The result is OK. The first half is done and if you said before the game we would win, I would take the result... Only the person who does not know about Hoffenheim can be surprised by this."
Young Boys dominated the second half against CSKA yet the Swiss ended up losing after an almost comical own goal.
A long ball was cleared into the Young Boys half which Nuhu headed into the air.
The Ghanaian then tried to head the ball back to goalkeeper David Von Ballmoos but instead it flew over the stranded keeper and bounced agonisingly inside the post and into the net as he tried desperately to claw it away.
Minutes earlier, CSKA goalkeeper Igor Akinfeev made a superb reflex effort to tip Guillaume Horau`s shot over the bar.
FCSB, European champions in 1986 in their old guise as Steaua, held out fairly comfortably against an uninspired Sporting despite having Mihai Pintilii sent off with 10 minutes left for a second bookable offence.
The Romanians, runners-up in their league last season, could even have snatched a win as Gabriel Enache and Junior Morais both went close in the second half.
Azeri champions Qarabag beat FC Copenhagen 1-0 and APOEL Nicosia were 2-0 winners at home to Slavia Prague in the evening`s other ties.