The Reds remain two points clear of Manchester City after a less-than-convincing 3-1 victory over the Clarets in front of the club's biggest league crowd of 60,725 following the full opening of the Anfield Road stand.
Diogo Jota had given the home side a 31st-minute lead, but after Dara O'Shea had levelled with a header in first-half stoppage time, it took goals from Diaz and Nunez, both created by substitute Harvey Elliott, to wrap up the points.
Reigning champions Manchester City has briefly replaced Liverpool at the top of the table after Erling Haaland's first goals since November finally wore down dogged Everton and claimed a hard-fought 2-0 win in the day's early kick-off.
The prolific Norway international broke the deadlock after 71 minutes of a drab encounter at the Etihad Stadium before wrapping up victory five minutes from time.
It was City's 10th successive win in all competitions and fired out another warning to their title rivals that their momentum is growing.
There was late drama in north London when Brennan Johnson struck in the sixth minute of stoppage time to fire Tottenham into the top four courtesy of a 2-1 victory over Brighton.
Pascal Gross' 17th-minute penalty had given the visitors a half-time lead, but the points looked set to be shared after Pape Sarr levelled in the 61st minute.
But Son Heung-min, on his return from Asian Cup duty, teed up fellow substitute Johnson deep into stoppage time to earn a vital win.
In-form Ivan Toney struck again as Brentford earned a deserved 2-0 victory at Wolves.
A third goal in four games since the striker's return from his ban for breaching betting regulations clinched the win at Molineux after Christian Norgaard's first-half header.
Bottom side Sheffield United cut the gap to safety to seven points with just their third Premier League win of the season at Luton.
Cameron Archer's strike and a James McAtee penalty handed the Blades a 2-0 half-time lead.
Carlton Morris' spot-kick reduced the deficit after the break, but Vinicius Souza ensured it ended 3-1 to the visitors.
Rodrigo Muniz scored twice as Fulham claimed their first league win of the year with a 3-1 victory over Bournemouth.
Muniz was given the nod ahead of January loan signing Armando Broja and repaid head coach Marco Silva's faith with two goals after Bobby De Cordova-Reid had opened the scoring, leaving Marcos Senesi's effort to count for little.
In the late kick-off, Bruno Guimaraes' double handed Newcastle a 3-2 victory at Nottingham Forest to make amends for their Boxing Day defeat in the reverse fixture.
The Brazil international's 10th-minute opener was cancelled out by Anthony Elanga.
Although the Magpies restored their lead through Fabian Schar, Eddie Howe's men needed a second from Guimaraes to claim just a third away win of the league campaign after Callum Hudson-Odoi had made it 2-2.
