It was the 57th minute when Núñez responded to Andersen's push by turning and headbutting the Dane in the face, leaving referee Paul Tierney with no option but to show a straight red card to the summer arrival from Benfica.
The hosts were 1-0 down at the time, after Zaha's superb finish rounded off a brilliant counter-attack in the 32nd minute.
Liverpool rallied in the minutes following the dismissal, though, and Díaz equalised with a sublime individual strike in the 61st minute to earn a draw that felt a fair reflection of the evening.
With two points from a possible six in their opening two Premier League games, the Reds will now be additionally keen to pick up three when they travel to face Manchester United next Monday, while Palace will want to build on what was an encouragingly balanced display when they host Aston Villa on Saturday.
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STORY OF THE MATCH
The injury-plagued Reds made four changes to the team that started their 2-2 draw at Fulham on the opening weekend.
Nat Phillips came in for Joël Matip in central defence, with the Cameroonian having sustained a groin issue since the trip to Craven Cottage, while Ibrahima Konaté remains out with a knee problem and Joe Gomez was only deemed fit enough for a place on the bench.
Harvey Elliott and James Milner took the places of Jordan Henderson and the injured Thiago Alcântara in midfield, and Roberto Firmino – also absent through injury – made way for Núñez up top.
Palace, meanwhile, made just the one alteration from their 2-0 home defeat to Arsenal 10 days earlier. That saw Joel Ward come in for Odsonne Édouard, as Vieira switched from 4-2-3-1 to a 5-4-1 shape out of possession, with Zaha leading the line.
On an idyllic Merseyside evening, Liverpool began with the kind of intensity that has set the tone for so many of their Anfield displays in recent times.
Within a minute, Milner had rifled an 18-yard left-footed attempt over after Eagles goalkeeper Vicente Guaita did well to push a Trent Alexander-Arnold cross away from Núñez.
Elliott, a lively presence in and out of possession, had the next opening in the ninth minute as he weaved away from challenges on the edge of box before his low – seemingly goalbound – shot was well blocked by Andersen.
Seconds later, Núñez volleyed a far-post Alexander-Arnold cross over.
Díaz was next to threaten in the 17th minute when he saw a deflected left-footed effort held by Guaita and, arguably, the best chance yet came 13 minutes later, as Mohamed Salah nodded a superbly flighted Alexander-Arnold pass into the gloves of Guaita.
That, as it turned out, was swiftly followed by Zaha's opener. The visitors sprang from their own third superbly before the excellent Eberechi Eze slid the 29-year-old through.
Zaha had timed his run perfectly, with replays showing that he was level with Phillips as the ball was played, before he sidefooted a superb finish across Alisson Becker and into the bottom-right corner via the post.
A brilliantly, ruthlessly executed sucker-punch.
The next chance fell to Zaha too, but this time Alisson raced out well to smother the effort.
There were a couple of sights of goal for the Reds prior to the break as Elliott first saw a 43rd-minute header fumbled behind by Guaita and Núñez then scuffed an attempt against the post after he was found by a cleverly flighted Elliott pass.
The second half began with more Liverpool pressure. Núñez was sent through by Salah but saw his left-footed attempt well blocked by Ward in the 48th minute – and the match's major flashpoint followed nine minutes later.
Andersen and Núñez had come together a couple of times in the box in the seconds prior to the red card offence, with the Liverpool striker seemingly moving his head towards the defender on two occasions as play progressed.
Then, after the ball went out of play, the Palace centre-back took issue with his opponent. His push of Núñez was followed by the 23-year-old turning and moving his head into the face of the Dane, who went to ground.
Referee Tierney, understandably, swiftly produced a straight red card. Andersen was also booked as a disappointed Núñez headed down the tunnel.
From there, though, the hosts channelled the spike in frustration superbly efficiently both on the pitch and in the stands and, four minutes after going down to 10 men, Díaz levelled in stunning fashion.
After being found on the left by Milner, the Colombian jinked between several challenges, worked his way into an inside-left position and fired a brilliant right-footed shot into the top-right corner in front of a delighted Kop.
Díaz roared and Anfield roared back. The intensity throughout the ground upped substantially in the moments that followed, with the goalscorer continuing to offer immense contributions in and out of possession, but Palace did well to limit clear openings from there.
They came close themselves in the 78th minute as Zaha diverted Cheick Doucouré's cross against the post on the stretch.
At the other end, Salah and substitute Fábio Carvalho each sent 18-yard left-footed volleys just wide in the 87th and 90th minute respectively – which represented the extent of the late openings.
A point that neither manager will likely be fully satisfied with, but one that simultaneously ought to offer several areas of encouragement.
Klopp may well be buoyed by the collective response that followed going down to 10 men, while Vieira ought to be very pleased with a performance that was disciplined, focused and dangerous on the counter.
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Núñez sees red
Much of the post-match focus will, understandably, be on the Uruguayan striker's actions. It was an immensely ill-judged move from Núñez that was rightly punished with a straight red. The aim will now, though, be to learn from such a scenario. It is a harsh lesson, but a potentially valuable one for the new No.27.
The response on the pitch and in the stands was a highly impressive one within the context of the evening. Díaz led the charge with a sublime goal and immensely energetic all-round display but Palace did well to settle back into the game and limit late opportunities.
Núñez now faces a three-match suspension, which will keep him out of the trip to Manchester United next Monday as well as home matches against AFC Bournemouth and Newcastle United.
Klopp hinted post-match that Firmino may be back in time for that trip to Old Trafford, which could be the go-to change within the front-three.
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PLAYER OF THE MATCH: EBERECHI EZE
Eze was a brilliantly classy presence for Palace. His assist for Zaha was the highlight, as he skipped beyond Fabinho and slipped his teammate through, but there was so much to go alongside that.
He was disciplined on the left of the midfield-four out of possession and consistently kept hold of the ball under significant pressure when the Eagles won it back.
A performance that offered a superb combination of focus and flair.