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A frustrating night for Jurgen Klopp's team appeared to be heading for defeat after Wilfried Zaha's first-half opener and Nunez's dismissal just before the hour.
Diaz's superb individual effort got his team level, though, before last season's Premier League runner-up set about trying to find a winner but to no avail.
It represents another early season blow to Liverpool's hopes of running champions Manchester City close again in the title race, already four points behind their rivals, but as with their opening draw against Fulham, it may feel like a point gained for Klopp in the circumstances.
Liverpool started brightly as they tried to find an early opener, with Nunez mishitting a volley at the far post, while Mohamed Salah also went close to scoring from Trent Alexander-Arnold's pull back, but the Egyptian’s side-footed attempt whizzed just wide of the near post.
After absorbing so much at the other end, it was Palace who took the lead in the 33rd minute when Eberechi Eze got away from Fabinho before playing Zaha in on goal, with the Ivory Coast international placing his shot past Alisson into the bottom-right corner.
Liverpool thought it was level just before the break as Harvey Elliott's chip found Nunez down the left side of the penalty area, but his deflected shot came back off the post.
Things went from bad to worse for Nunez as the Uruguayan reacted to being shoved by Joachim Andersen by pushing his head into the Dane, with referee Paul Tierney immediately producing a red card for the Liverpool striker.
However, the Anfield atmosphere soon improved as Diaz dribbled inside several Palace defenders before unleashing a rocket of a shot past Vicente Guaita's despairing dive to his left.
The host tried to find a winner despite having fewer players, though Palace could also have won it when Zaha missed a great chance at the far post, but both were forced to settle for a point.