Roberto Firmino stepped off the bench to score a late leveller as leaders Arsenal squandered a two-goal lead, and the opportunity to move eight points clear of Manchester City at the top of the table.
Mikel Arteta’s side went ahead on eight minutes when Gabriel Martinelli poked the ball past Alisson for his seventh goal in eight Premier League matches after good work by Bukayo Saka and Martin Odegaard.
Jesus, who had never previously lost a Premier League match in which he has scored, headed in from Martinelli's left-wing cross for his third goal in two matches since returning from injury.
Salah was handed a great chance to haul Liverpool level from the penalty spot on 54 minutes after Rob Holding fouled Diogo Jota, but the Egyptian fired wide from 12 yards to make it consecutive misses from the spot, following his failure in a 1-0 defeat at AFC Bournemouth.
Liverpool kept the pressure on, and Aaron Ramsdale came to Arsenal's rescue with a crucial save from substitute Darwin Nunez as the match entered the final 10 minutes.
But moments after Gabriel headed straight into Alisson's arms, play swung the other end and Liverpool were level with Firmino's looping header from Alexander-Arnold's delivery. It was Firmino's eighth goal in eight matches against Arsenal at Anfield.
Ramsdale then produced breathtaking saves to deny both Salah and Ibrahima Konate in a frenetic finish to a pulsating encounter.
Arsenal remain top with 73 points, six clear of Man City, while Liverpool stay eighth on 44 points.
Palace in five-star display at Leeds
Leeds started on the front foot and Patrick Bamford had already clipped a post before he opened the scoring on 21 minutes with a glancing header from Brenden Aaronson's corner for his 50th goal for the club in all competitions.
Palace were indebted to debutant goalkeeper Sam Johnstone for denying Leeds further goals as he brilliantly tipped-over Jack Harrison's free-kick before saving Robin Koch's header from the resulting corner.
Leeds had 10 shots, their most in the first half of a Premier League match this season, but it was Palace who drew level in added time when Marc Guehi nudged the ball past Illan Meslier for his first goal of 2022/23.
Palace had won a competition-high 17 Premier League points from a losing position this season, including a 2-1 win in the reverse fixture against Leeds, and a blistering start to the second half put them on their way to making it 20.
He then completed his hat-trick of assists for Odsonne Edouard to add a fourth on 69 minutes before Ayew netted his second with 13 minutes remaining to secure Palace's sixth comeback win of the campaign.
Aged 21 years and 118 days, Olise is the youngest player in Premier League history to assist three goals from open play in a single match.
He now also has 13 top-flight assists since the start of last season. The only player currently aged 21 or younger with more in the competition during this time is Saka, with 17.
Back-to-back wins take 12th-placed Palace to 33 points, six clear of the bottom three. Leeds slip to 16th, only two points off third-bottom Nottingham Forest.