In what was a one-sided affair at Turf Moor, the Reds emerged as 2-0 winners to go two points clear in the Premier League.
It took just six minutes for the Merseyside outfit to take the lead, with Nunez grabbing his first top-flight goal since the 3-0 victory over Nottingham Forest on October 29.
The Uruguay international was picked out on the edge of the box by Cody Gakpo, with the 24-year-old's first time curling strike nestling into the bottom right corner.
Liverpool's opener should have been the first of many, but they spurned a number of chances - with Mohamed Salah smacking the crossbar when sent clean through.
Jurgen Klopp's men did find the back of the net before the break through Gakpo, but the goal was ruled out after Nunez was adjudged to have pushed Charlie Taylor.
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It was more of the same in the second, with the visitors again having a goal ruled out by VAR.
This time it was Harvey Elliott who thought he had doubled his side's lead, with the England youngster stabbing home from close range.
However, Salah, who was in an offside position, was determined to be in front of James Trafford's eyeline - hindering the Clarets goalkeeper's vision.
Against the run of play, the hosts almost managed to find an equaliser through Johann Berg Gudmundsson.
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Just eight minutes after coming on, the Icelandic forward headed over the bar from just a few yards out with Alisson stranded.
And Vincent Kompany's side were made to pay, with Diogo Jota getting on the score sheet in the final stages in what is his 50th Liverpool goal.
A number of Reds players led a counter-attack in the 90th minute, with Jota firing through Trafford's legs from an acute angle.
With Liverpool to host Newcastle on New Year's Day, they knew they couldn't afford to drop more points following back-to-back draws against Manchester United and Arsenal.
The Gunners have the chance to reclaim top spot when they entertain West Ham on Thursday night.
As for Burnley, they remain 19th in the table - two points above Sheffield United.