The Reds travelled to the Brewers' Pirelli Stadium home for the first ever meeting between the two clubs, but the gulf in class was obvious from the outset.
Divock Origi's neat flick broke the deadlock after Sadio Mane's set-up, before a Roberto Firmino header from Nathaniel Clyne's cross doubled the lead before half-time.
After the break, Klopp's visitors ran rampant as a Tom Naylor own goal, in addition to Daniel Sturridge's double, put the gloss on a convincing victory to secure passage to the third round.
The manager felt that it was the manner in which his players put the lessons learned from Saturday's 2-0 loss to the newly-promoted Clarets which helped them to sweep aside their Championship opponents.
Reds boss: Our display made sure no upset was possible
Klopp told journalists post-match that he noticed "a few situations" where his side "missed a little sharpness" at Burnley, but noted that their "main problem" was that "with one more pass" they would have been "15 times completely free in the box."
He suggested that his players "didn't see" that final pass in the game, but could in their analysis afterwards, and spoke about how they "needed to use this", adding: "That's what we did today and it was better. We were there, where we had to be."
Klopp explained that his side "were on the line", had "possession [in and around] the box" and got "bodies in the box" as he said that the "first two goals" were examples of their improvement.
"One was a transition, the other one after a second ball," the German continued, calling it "all good" as he said that even after a loss, there are "a few things" that you "have to use" to improve.
The Reds boss was pleased with his side's performance, declaring that they made sure it was "not as difficult" as they "could have imagined or expected it to be" against Nigel Clough's second-tier side.
He added that Burton was "the most difficult draw" they could have had, being away from home at one of the highest-ranked sides in the draw's hat, and said that it was "possible" an upset could have occured.
Klopp therefore praised his side for being "very professional" and "really serious" in their approach, hailing them for doing their job collectively and saying "it was good."
Klopp hoping Origi and Sturridge can continue to do battle for single striker role
Klopp was inevitably asked about the performance of Sadio Mane, the winger returning from a shoulder injury to play a part in four of their five goals and directly assist two.
The winger didn't give Burton's defence a moment's rest with his pace, movement and sublime passing, but Klopp insisted he doesn't want to rain praise on the Senegal international.
"I like this player but I do not like talking about him too much," he said, with many having suggested Mane's absence played a huge role in the weekend's Premier League loss at Turf Moor.
Klopp declared that they "do not have to celebrate ourselves every day" just because they signed Mane and focused on the need to "try and improve" before saying that there is "nothing else to say about this at the moment."
Among the others to impress were Origi and Sturridge, both of whom found the back of the net for the first time this season.
The manager was pleased with what he saw from the two strikers and hopes they can continue to play so well in order to hand him some tough selection decisions for future fixtures.
Klopp said that they "have to make the decisions" but insisted his job is "to find the right [decisions] to make the right solutions."
"The job the players have to do is to be in the best shape they can be in," he said, calling it "clear what both sides have to do" and hoping that he will have "difficult decisions" to make with his line-ups moving forward.