The Merseyside giants made it seven Boxing Day victories in a row against Vincent Kompany's charges, who succumbed to strikes from Darwin Nunez and Diogo Jota either side of the break.
Nunez's composed finish found the far corner with just six minutes on the clock, but Liverpool failed to convert their superiority into a bigger first-half margin, as Cody Gakpo had a goal disallowed while James Trafford made seven saves in the opening 45.
Nunez was adjudged to have felled Charlie Taylor for Gakpo's chalked-off effort, and Elliott also fell victim to the officials in the second half, seeing a sublime first-time finish ruled out for Mohamed Salah being in an offside position.
Salah was ostensibly pushed into Trafford's path, but the VAR room determined that he was still interfering with play from an offside position, and a galvanised Burnley proceeded to ask a few more questions of the Liverpool backline.
The Reds emerged unscathed, though, and they were finally out of sight in the 90th minute, when Jota combined with Luis Diaz and managed to fire through Trafford's legs from a tight angle in the left-hand side of the area.
Liverpool's success means that they will temporarily sit at the summit of the Premier League table, at least until Arsenal return to action on Thursday, but he was less than impressed with the decision to disallow Elliott's effort.
Speaking to Amazon Prime, the German said: "Only somebody who has never played football can make this an offside. It's insane when you see that. Why do referees create their own view on something?
"We all watch football and need the referee but we need them to make the right interpretations of the rules. It's like, wow. Handball is sometimes handball and all these kinds of things."
Nevertheless, Klopp was able to revel in an "incredible" performance from his charges at Turf Moor, although he acknowledged that his side have been lacking a clinical touch in recent fixtures.
"We tried to find what was the Liverpool way and it looks a little bit like throwing hurdles in front of us and finding ways to get over it," Klopp added.
"We played an incredible game and then it was just 1-0 which was really strange! That's the way it is in football. With the most difficult situation in the whole game probably, we make it 2-0.
"We've created a lot of chances recently and haven't used an awful lot of them but in the end it's about performing. We've played four games in 10 days and you just have to get through it. In general, the game we played was exactly how we wanted to play.
" In that time they had no solutions but we left the door open and that's how it is. Now we have five days between now and the next game so it feels like we should go on holiday? No, the boys will have two days off with the family."
Liverpool's beating of Burnley represented their final fixture of 2023, and the Reds will open the New Year with a home clash against Newcastle United at Anfield on January 1.