The Merseyside giants had squandered the chance to seal qualification for the knockout stages on matchday four in Group E with a 3-2 loss to Toulouse, who sat just two points behind Klopp's men heading into the penultimate round of fixtures.
While a win would have therefore guaranteed Liverpool a top-two finish with a game to spare, victory for Toulouse over Union SG meant that the Reds would have to wait until the final matchday to learn whether they would directly enter the last 16 or face a third-placed Champions League team in the playoffs.
Klopp's men did all they could to confirm top spot with a four-goal thumping of the Austrian side, who courageously ventured forward in numbers at Anfield but were thwarted by a steadfast Reds backline.
Twenty-year-old Quansah made his third Europa League start of the season and garnered high praise for his performance, completing 91% of his passes and making three clearances to help preserve Liverpool's clean sheet.
Speaking on Quansah's display at the back, Klopp waxed lyrical about the England Under-21 international's "spectacular" work throughout the evening and pointed to his development as the reason for Liverpool not pursuing a new centre-back over the summer.
"Look, for us that's absolutely perfect. Jarell played a super game. That couldn't be better news for the club, to be honest. I think before this season a lot of people told us we should buy a centre-half," Klopp told his post-match press conference.
"We knew we had Jarell. Did we know that he will show up like this? No, not exactly, but we were pretty hopeful, so the future is bright. It's cool.
"Our own boy, calm on the ball, good in the challenge, quick, ruthless in the right moments and besides Caoimhin [Kelleher], who had one or two saves, [he] probably had the most spectacular defensive situations when he saved our backside in some moments. So, that's really good. [It's] top news for us."
A few minutes after the final whistle had blown at Anfield, Toulouse's clash with Union SG ended in a goalless draw, confirming Liverpool's place in the last 16 as Group E winners with one match left to go.
With the Reds also fighting in the Premier League, EFL Cup and FA Cup heading into the New Year, Klopp labelled early qualification "very important", adding: "Two top results for us tonight in the group.
"It was clear, with the defeat at Toulouse we made it a bit more tricky, but because of our result and their result [tonight] we are now top of the table and that will not change. That's good, very important in the busy, busy, busy schedule we have from now on. [There were] a lot of positives in the game, a lot of good football.
"The thing I didn't like too much, and told the boys at half-time, [was] this game should have been put to bed already at half-time. A third, fourth, fifth goal... you cannot play better, you cannot set it up better, [but] you can finish it better obviously, but we didn't.
"So, what was it? Fifty-something [minutes] when we scored the third? That's then very helpful as well. Scored a fourth goal in the last minute or whatever, so result top, performance really good [and] nobody got injured. All good."
Liverpool have two full days to recuperate from Thursday's triumph - which was their 10th in 10 home matches this season - before Fulham visit Anfield for Sunday's Premier League affair.