Darwin Nunez atoned for Liverpool's first-half errors and clinched a remarkable three points at Newcastle
Liverpool were battling against the odds and second best for much of a thrilling encounter, falling behind to Anthony Gordon's 25th-minute goal after he cashed in on Trent Alexander-Arnold's mistake, then seeing captain Virgil van Dijk sent off for a foul on Alexander Isak three minutes later.
Alisson kept Liverpool in it by miraculously turning Miguel Almiron's shot on to the woodwork in the first half, the forward also hitting the post after the break, before Uruguay striker Nunez was introduced to stunning effect.
Nunez ran through to rifle a shot past Newcastle keeper Nick Pope with nine minutes left, then took Mohamed Salah's pass in his stride three minutes into stoppage time before applying an equally emphatic finish to give Liverpool an astonishing win.
Liverpool's win was a triumph for strength of character and stickability because they spent so much of this match outnumbered, out of sorts and clinging on before they suddenly clicked into gear and took the points.
The starting point must be goalkeeper Alisson, who produced that remarkable save from Almiron when Liverpool were 1-0 down, somehow turning his powerful shot on to the frame of the goal. All reality suggests even Jurgen Klopp's side would not have come back from that.
Liverpool shifted the emphasis of the game with the introduction of substitutes Diogo Jota and Nunez, positive moves in a very negative situation, suddenly giving them a threat when before it had been merely about survival.
Nunez has been very much hit and miss since his big-money move from Benfica, but he was all hit here on Tyneside, that first explosive finish sending shockwaves through St James' Park, the stadium suddenly rocking with nerves rather than celebration.
Just as a point looked like Liverpool's reward, Nunez delivered again to send Klopp - who had spent so much of the afternoon frustrated and angry - wild with celebration on the touchline at the end of this eventful afternoon.
Newcastle manager Eddie Howe and his players looked almost bewildered as the final whistle sounded after a rollercoaster encounter and Liverpool were celebrating a superb smash-and-grab win.
They were left to rue failing to take advantage when they were in complete command against Liverpool, reduced to ten men, a goal behind and struggling to cope with the intensity of Newcastle's pressing game.
Alisson was the barrier, while the woodwork also stopped Newcastle extending the lead given to them by Gordon's smooth finish - chances missed and opportunities squandered before they paid a heavy price.
There was also carelessness when substitute Harvey Barnes selfishly went alone with Callum Wilson unmarked in front of goal with the score 1-0 - and with Liverpool's array of attacking talent the danger always lurks, as Newcastle found to their cost.
It made it two defeats from Newcastle's first three games after the opening-weekend thrashing of Aston Villa, and this one will cause real pain because they will know they should have had this one wrapped up before Nunez's late double.
-- Courtesy of BBC Sport