The Reds moved into the top four with a dominant display at the Vitality Stadium and led 3-0 by half-time thanks to Philippe Coutinho's solo effort, a Dejan Lovren diving header and Mohamed Salah's composed finish.
But at the same stadium in which they twice coughed up two-goal leads on their previous visit last December, Liverpool showed no signs of complacency and added a fourth through Roberto Firmino.
Liverpool are now unbeaten in 12 matches in all competitions and became the first team in top-flight history to win four successive away games by at least three goals.
Klopp delighted by "fantastic" goals and controlled display
Speaking to journalists afterwards, Klopp declared himself "very" satisfied afterwards, insisting his mood is "how you can imagine" and noted that Bournemouth is "a really tough place to come."
He insisted that without the concentration or focus that Liverpool showed against the Cherries "it's impossible to get something here" but hailed his charges for being "from the first second really, really spot on."
"I liked the mix-up of being patient in different moments and being very lively in others. Very good organisation, defending, approach, idea and it worked out," the German explained.
He felt that it was "clear" Liverpool could not "create chances minute-by-minute" but hailed "a fantastic first goal", insisting he wanted to give "40 per-cent to Andy Robertson."
Klopp insisted: "If he doesn't make the run in Phil’s back then I think this version is completely different, but of course it was fantastic of Phil."
The Reds boss said that Firmino "forced the second goal" and praised Liverpool's "fantastic third goal" though he admitted it "looked a little bit like offside" and also "a fantastic fourth goal."
Klopp insisted that Liverpool "controlled the game in the second half" and "came not really in trouble" because Bournemouth "had only one real chance" while the away side "had a lot more than we scored."
"I think it’s pretty fair that they didn't use the one [chance they had]," reflected Klopp. "That's happened really often against us [teams taking just one chance]. So I am really happy about the full package we showed. It was difficult, so we are really satisfied."
The win lifts Liverpool back above Arsenal - who they face in North London on Friday night - into the top four, though Klopp admitted it was not something his side focused on.
"We couldn't be higher," he acknowledged. "Since this morning it was clear that if we win we are fourth, so yes I am happy about it but I didn't think about it, to be honest."
Klopp hails importance of self-assured Salah
Salah became the joint-second quickest player to score 20 goals for Liverpool, having done so in the same amount of games (26) as team-mate Daniel Sturridge.
Only George Allan, who scored 20 goals in 19 matches back in 1895, has done so quicker than Salah - one of the early candidates for the Premier League's Player of the Season award.
Asked about the Egyptian's phenomenal goalscoring form this season, Klopp insisted that even if he failed to score "from now on" then it would be "still 20 after the season", hinting at the incredible success that Salah has enjoyed.
The manager said that he is "looking forward" to Salah staying "in that shape" and continuing to play "like this", acknowledging that the player himself "knows how much talk" there is about him.
Klopp added: "He knows how much I respect and like him, but it's not about the 'Fab Four', and I don't like that name as well. I don't go home and think 'My God, what a player Mo Salah is, how good is he and all the others?' That's not how it works."
However he noted that the BBC African Footballer of the Year is "a very important player" for Liverpool, "like the others", and insisted "in the moment" it is easier for Salah "to finish the situations than the other players" due to his form and confidence.
"Sadio [Mané], [Alex] Oxlade-Chamberlain, have big potential in this. Adam Lallana is back, so we need to bring altogether that we can play intensive football. Not because we want to play intensive football, only because the period is so intense that you have to cope with it, we have to deal with the situation. For that we need all the players," he said.
"Liverpool prevented Cherries from playing their usual game"
Klopp declared himself happy at his side "more importantly" claiming three points and asked to surmise Bournemouth's display, simply felt that Liverpool "were difficult to play."
He revealed that their pre-match analysis showed Eddie Howe's side were "well organised" and had a "clear structure" but said Bournemouth did not show their "real face" and Liverpool "showed what we can do and how we can cause them problems."
Klopp insisted that Sunday's match did not see "the things you see normally in a game from Bournemouth" but "mainly" because of Liverpool's "really good" performance.
"We were good in situations, everyone ready for defending," he lauded. "We had a defensive plan and it worked out, and that's good."