The Reds were 1-0 down early on after Jonathan Walters' header beat Simon Mignolet at his near post and were fortunate to the Belgian for denying former Liverpool midfielder Joe Allen shortly after.
That triggered a much-needed response through Adam Lallana and Roberto Firmino, as the hosts went into half-time ahead.
Liverpool were comprehensively the better side after the break, Giannelli Imbula netting an own goal from Divock Origi's driven delivery across the six-yard box.
And substitute Daniel Sturridge marked his arrival with a goal inside a minute to seal an excellent three points for Klopp and co. by rounding the goalkeeper and sliding in for his first league goal in nine months.
Reds boss: First two goals were "really good and really important"
Klopp, speaking in his post-match press conference, declared that confidence "is not a problem of the team at the moment" and that they "know about our quality" although they "don't show it all the time."
"The start of the game was really difficult because the plan of Stoke was obviously to press really high, especially with the two strikers," continued the German.
He lamented that his side "were not patient enough in our passing game in these moments" and noted that while they were under pressure, they "could have done much better" and allowed it to become what he described as "a wild game."
Klopp felt that former Reds striker Peter Crouch was "outstanding" as he said that whenever Stoke had possession, the ball "was immediately in the air" and it was "really difficult to defend."
He also praised how Allen was "brilliant in the beginning" in winning the second balls for Stoke, ensuring Liverpool had "to work really hard" in a "completely open" game.
The manager continued: "In creating our moments, we again struggled a little bit with [defensive] protection, with our position game. A lot of things we could have done better, but we forced two goals with our quality. They were really good and both really important."
Klopp insisted that, at 2-1 up thanks to Lallana and Firmino's goals, they "could again adjust a little bit at half-time" and "speak about a few things."
He felt his players "did better" and "scored a wonderful third goal" before Sturridge "closed the game" which was "really important" - adding that it "was clear" that Stoke "are a good team" and that Mark Hughes "likes to bring Crouch against Liverpool" because he is "always involved."
Klopp "more than pleased" for Sturridge and hails Henderson and Sturridge importance
Klopp raved about the contributions, and the importance, of players such as captain Jordan Henderson and goalscorers Lallana and Sturridge.
The Reds boss declared that Henderson is "a very important player" but insisted it is "not the moment to praise single players" because they don't need it "in this moment" and they "are all happy about the team performance, the result and the goals we scored."
He continued: "He [Henderson] has been doing really well over the long-term, but it's not a moment to lean back or whatever. He will not. But my feeling is always that when we speak about one player, we speak always about the past."
Klopp declared that he is already thinking "about the next game" and hoped that Liverpool "can do better" against their next opponents, Manchester City, because they "can do better than we did tonight" although it was "absolutely okay" for them to beat the Potters.
On Lallana, Klopp continued that if only the midfielder "was good at keeping the ball, then we have a big problem" but added: "I saw no player who didn't do his job tonight. It was not easy."
He reiterated that Liverpool "could have done better" and "especially in the last 15 minutes" when they were "4-1 up" and "ran with the ball still on fire" which he felt "makes not too much sense" with such a commanding advantage.
Yet he acknowledged that their all-action displays are "part of our character" and so insisted he doesn't have "too big a problem" with their intensity.
Klopp insisted that they "could have done a lot better", the entire squad, in the first-half but praised Lallana for being "very important."
He also said he was "more than pleased" with Sturridge's goal, his first since netting in a home draw against Newcastle United in April, calling it "a perfect goal."
"In a situation like this, 3-1 up but still intense, and then you can use the quality Daniel has, that's perfect," Klopp stated, insisting that "doing what he did [rounding the goalkeeper to score] was perfect."
He labelled it "a wonderful goal" which was "very, very important" for them because it "closed the game" and "gave us the opportunity to cool down a little bit."
Man City fixture a "special game", says Klopp
Klopp was inevitably asked about what awaits next for his Liverpool side, who host Pep Guardiola's City on New Years' Eve in their final game of the calendar year.
He acknowledged that it is a big game "for both" teams and suggested that Guardiola being at Anfield to watch their win over Stoke was the "first sign" that Saturday's encounter is "a special game" because he hasn't "watched a lot of games in the last few weeks from the stadiums of the opponents."
Klopp declared that his side are "already looked forward to" what promises to be "a difficult game for both teams" but one that will be "exciting."
He insisted that "the best thing" is that the game is "at Anfield", adding: "I'm really looking forward to it. They are an outstandingly good side and we are not too bad, so it will be a nice game."
Klopp also joked that he hoped Spaniard Guardiola, who managed enjoyed great success with Bayern Munich when he was Borussia Dortmund boss, "saw nothing" because he "had to write all the autographs."
He said that visiting rival's grounds is "more a signing hour than you can watch a game", concluding: "The game is on December 31st and whatever I say tonight can't win it. But maybe I could say a few things that would make it a bit more difficult for us to win it, so it's probably best I shut my mouth!"