The Reds travelled to Amsterdam needing only a point to confirm their place in the next round, but they initially struggled to impose themselves on the game.
Setting up in an unfamiliar diamond formation, Liverpool were on the back foot for a good chunk of the first half before Mohamed Salah beat an onrushing Remko Pasveer to break the deadlock.
Darwin Nunez then hit the post from close range before making amends with a header to double his side's lead in the second half, and Harvey Elliott finished from a tight angle to wrap up the three points.
Liverpool now host Napoli next week safe in the knowledge that they will be in the pot for the last-16 draw, but Klopp does not want his side to take their progress in Europe as a given.
"Qualifying for the knockout stages, I think nobody ever should take for granted - it's really difficult. We did it again, which is a big achievement in itself," Klopp said in his post-game press conference.
"In the game, it started really, really tricky for us - and that makes the victory in the end even sweeter because Ajax are a really good team and we've beaten them twice. It was very important.
"Today, the first half an hour, maybe 35 minutes, they were the better team. They looked much more settled than us, but that might be normal because we changed again slightly and from our point of view we had to.
"I'm not sure what is better in our situation, to fly through a game - probably everybody would love it - [or] to overcome difficulties and in the end to develop into your best self is obviously very helpful as well. Today it was like that.
"We scored wonderful goals, like really wonderful goals. The first goal, the overview of Hendo was exceptional. The second goal was a great cross and the whole situation how Darwin takes on pretty much everybody. And then Harvey crowns his really good performance with a sensational finish."
The jubilant mood briefly turned to one of concern for Liverpool when Jordan Henderson had to be replaced in the 71st minute after a knock, with James Milner coming on for his countryman.
However, Klopp is hopeful that the midfielder did not suffer more than a painful bruise, and the rest of his team also came through the progress-clinching win unscathed.
When asked if Henderson had avoided a serious injury, Klopp responded: "I think so. I was worried as well when I saw the situation because it was in front of me, but I think he thought he got knee on knee, so painful.
"A bruise, probably. Really painful but as far as I know now, nothing else and that should be fine. I didn't have too much time to spend in the dressing room yet, but no-one came to me with any further or bigger problems."
Liverpool remain three points behind a perfect Napoli in the Group A rankings and must beat the Partenopei by a four-goal margin or more to pip them to top spot on the final matchday.
The Reds will now aim to return to winning ways in the Premier League when they tackle Leeds United at Anfield on Saturday evening.