Diaz met Trent Alexander-Arnold's beautiful cross with a diving header to open the scoring before Alexis Mac Allister nodded home from close range to double Liverpool's lead.
James Maddison pulled one back four minutes before half-time after Mac Allister was dispossessed in midfield by Dejan Kulusevski to give the hosts a route back into the contest.
But Dominik Szoboszlai scored in first half added time to restore the Reds' two-goal lead and give Spurs a mountain to climb.
Salah netted twice in seven second-half minutes, passing Liverpool great Billy Liddell on the club's all-time scoring list with his 14th and 15th league goals of the season, to make it 5-1 and send some Spurs fans heading to the exits.
"I’m happy and proud, I’ll just keep working hard," said Liverpool's now fourth most prolific scorer.
Dejan Kulusevski's volley was followed by Dominic Solanke's 83rd-minute goal to give Spurs hope of an unlikely comeback, before Diaz made it 6-3 two minutes later and finish a madcap contest that could have had even more goals.
"We were quite good up front but defensively we need to improve as a team," added Salah to Sky TV. "It’s quite good the result, and hopefully we just keep going."
Liverpool are top with 39 points from 16 games, four points above second-placed Chelsea who have played a game more. Spurs remain 11th, with 23 points from 17 games.
Breathless
The Reds constantly threatened in attack and Salah ought to have put them ahead inside three minutes when Spurs goalkeeper Fraser Forster, who made two costly errors in midweek, passed straight to him in the box, but Salah dragged his shot wide.
Salah then hit the bar after beating three defenders with some silky footwork, but the visitors deservedly took the lead in the 23rd minute with Diaz heading home at full stretch.
Mac Allister was at fault for Spurs' first goal, however, which came out of nowhere and ought to have given Spurs a platform to drag themselves back into it.
But Szoboszlai made it 3-1, running onto Salah's through ball and nutmegging Forster at the end of a sweeping move.
Salah got in on the act after the break, tapping home from close range in the 54th minute and then adding another shortly after when Szoboszlai laid it on a plate.
Liverpool showed no signs of sitting back, with Forster forced into a good save to deny Alexander-Arnold, before Kulusevski and then Solanke raised the prospect of a remarkable comeback.
But Diaz placed a clinical finish past Forster with five minutes of normal time remaining to end a breathless encounter, which fell two goals short of equalling a Premier League record.
Man United in bottom half at Christmas for first time
Bournemouth thrashed Manchester United 3-0 in the Premier League at a chilly Old Trafford on Sunday and climbed to fifth in the table thanks to goals from Dean Huijsen, Justin Kluivert and Antoine Semenyo.
Bournemouth's fourth league victory in five matches moved them to 28 points after 17 games, while United are 13th with 22 points and will spend Christmas in the bottom half of the table for the first time in Premier League history.
"I always say in this kind of situation we have to show personality and character," United defender Lisandro Martinez said. "We are playing so well and unfortunately, we couldn't score today. We have to believe. At this club, we have to win every game and we know that."
While United peppered Bournemouth with shots late in the first half, including three from captain Bruno Fernandes, it was the visitors who put the game away with two goals within two minutes early in the second half.
Kluivert scored from the penalty spot in the 61st minute - awarded after Noussair Mazraoui brought down the Dutch winger - sending keeper Andre Onana the wrong way.
The delighted Bournemouth fans were still celebrating when Dango Ouattara swept a pass into Semenyo inside the box two minutes later. Martinez put up little resistance as Semenyo unleashed a hard shot past Onana.
"It's an important win for us," Bournemouth manager Andoni Iraola said. "We won 3-0 but it was more difficult than it looks when you see 3-0. I think we've been quite solid, it was a mature performance."
'Man United, it's happened again'
Bournemouth's supporters gleefully sang "Man United, it's happened again" in the dying minutes, in reference to the Cherries' victory by the same score at Old Trafford last season.
"Feels great, back-to-back wins at Old Trafford," Semenyo said. "We know we have a good team and we are taking it game-by-game. I am not going to say too much and I don't want to jinx it."
United looked completely out of sorts at both ends of the pitch for most of a game that came on the heels of their feel-good 2-1 derby victory over Manchester City a week earlier.
"It's hard to push for two, three wins and we are trying," United boss Ruben Amorim said. "This game was hard on us. We suffered again on set-pieces and we were a bit nervous, I felt it in the stadium. The penalty and another goal are really hard here. We tried to score some goals but it was a tough match so let's move on.
"We have to focus on the job and not what you feel in the stadium. It's the only way I know how to focus my players. We have to suffer again but we will try to win. We will do it until the end."
United had 23 shots to Bournemouth's 10, including the flurry right before the break that saw Fernandes fire a rocket from the top of the 18-yard box that sailed just wide of the net. Seconds later, Kobbie Mainoo beat three defenders before shooting just wide of the post.
Alejandro Garnacho squandered a terrific chance in the second half when he outsprinted two defenders before shooting at keeper Kepa.
Amorim left forward Marcus Rashford out of his squad for the third successive match, after rumours his time with the club could be limited.
The 27-year-old forward, who was at the stadium on Sunday but not on the bench, appears to have fallen out of favour, and said in an interview earlier in the week that he was ready for a "new challenge" away from the club.
