Having survived a number of attempts on goal by the hosts, which included Steven Berghuis rattling the woodwork early on, the Reds went ahead in the 42nd minute at the Johan Cruyff Arena via Mohamed Salah's finish.
Darwin Nunez hit a post soon after, before extending the visitors' lead with a 49th-minute header, and Harvey Elliott then added a fine strike three minutes later as Klopp's men, needing only a point to join Group A leaders Napoli in the next round, claimed all three.
Liverpool conclude their group games next Tuesday by hosting Napoli, who they are three points behind and lost 4-1 to in their European campaign opener.
Klopp said: "I didn't think a second about the Napoli game yet, but 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, in a game which started off really tricky for us, and that makes the victory in the end even sweeter, because Ajax are a really good team, and we beat them twice (also winning 2-1 at Anfield in September).
"Today the first half an hour, maybe 35 minutes, they were the better team. They looked much more settled than us - that might be normal because we changed again slightly, and we had to press slightly different and had to get used to that. But that's normal, it's an away game in the Champions League, it's difficult.
"We came through that, with one situation where we were probably lucky and the other with a good block and then we scored our goal, and you can see how the goal can open pretty much everything.
"It was brilliant and the next situation was a sensational football moment, unfortunately we hit the post there. Then we scored the other two goals and controlled the game, which is really good, and now we are really happy about going to the last 16."
Klopp's side returned to winning ways four days on from the shock 1-0 Premier League loss at Nottingham Forest.
The German added: "I'm not sure what is better in our situation, to fly through a game or to overcome difficulties and in the end to develop into your best self - that is obviously very helpful as well, and today it was like that.
"In the end we just did really well. I would be over the moon if we played here, 0-0 and got through, because it is all about getting through, and we did that with the 3-0, and that's perfect.
"We scored really wonderful goals, the first goal, the pass by Hendo (Jordan Henderson, for Salah) was exceptional, the second goal was a great cross (by Andy Robertson) and Darwin takes on pretty much everybody, and then Harvey crowned his really good performance with a sensational finish."
Skipper Henderson came off in the 71st minute after hurting his knee.
And Klopp - currently without a number of players due to injury - said when asked if the midfielder was OK: "I think so. I was worried as well when I saw the situation, because it was in front of me. I think he got knee on knee, so painful, a bruise, but as far as I know now nothing else."
With their last-16 hopes over, third-placed Ajax will now look to secure Europa League football when they play Rangers away next week.
Boss Alfred Schreuder told the club's official website: "We played well in the first 35, 40 minutes. One of the chances has to go in.
"Some things went wrong at 0-1, at 0-2 we give away a corner unnecessarily and the boy (Nunez) is not allowed to head in so freely.
"The biggest difference with Liverpool was that there was more guts. You saw that it was rewarded."