Salah, fresh from his six-minute hat-trick against Rangers, produced a brilliant piece of control to take down Alisson's long kick and spin away from Joao Cancelo before beating Ederson with a cool finish (min 76.) as Liverpool put their recent struggles behind them.
Klopp's red card came as tempers frayed in the closing stages, the Liverpool manager dismissed after reacting furiously when the hosts were denied a free-kick for an apparent foul by Bernardo Silva on Salah, whose goal ended a five-game Premier League scoreless streak.
That incident came after Pep Guardiola had also been left irate earlier in the game when a Phil Foden goal was overturned by VAR after Erling Haaland was deemed to have fouled Fabinho in the build-up.
Liverpool came into the game 13 points behind City following their worst start to a season in 10 years but they defended resolutely, becoming only the second Premier League side all season to prevent Haaland from scoring, and seizing their chance when Salah, who had earlier been denied by Ederson in similar circumstances, caused Anfield to erupt with a goal which breathes life into their season.
For City, meanwhile, the defeat brings an end to their unbeaten start to the Premier League campaign and leaves them four points behind early leaders Arsenal, who beat Leeds 1-0 earlier on Sunday.
Liverpool were facing a momentous task against a City side who had scored 13 goals in their last three Premier League games and that task was made even tougher before a ball had been kicked.
A muscular injury to Ibrahima Kante saw the French international join Joel Matip on the sidelines, meaning Joe Gomez had to move across to centre-back and James Milner had to fill in at right-back, with Trent Alexander-Arnold only fit enough for the bench.
The makeshift backline faced a stern test against the free-scoring Halaand, and the City striker had a series of half-chances before the break fresh from being rested against FC Copenhagen in midweek.
The first of those chances saw him denied by Alisson when he attempted to chip the Liverpool goalkeeper from Ilkay Gundogan's pass. He then twice spurned headed opportunities, the second of which came from a trademark Kevin De Bruyne cross.
Liverpool held firm, though, and created openings of their own in the first period too, with Diogo Jota sending a free header straight at Ederson from Harvey Elliott's cross, and the returning Andrew Robertson blazing over following another dangerous attack.
It was after the break, though, that the game truly exploded into life.
A frenetic period began with Ederson brilliantly saving from Salah after the Egyptian had sprung City's offside trap from Roberto Firmino's pass, and within minutes City thought they were in front.
Foden was the man to have the ball in the net, firing in via a deflection off the otherwise outstanding Gomez after Alisson had failed to gather a De Bruyne pass under pressure from Haaland.
But City's joy was short-lived, with referee Anthony Taylor instructed to consult the pitch-side monitor and duly penalising Haaland for a shirt pull on Liverpool midfielder Fabinho.
The reprieve emboldened Liverpool, with Jota going close again when he headed over from a Salah cross at the far post, but City continued to threaten too, with Haaland during a superb one-handed save from Alisson soon afterwards.
It was Liverpool, though, who made the decisive breakthrough, Salah's outstanding control from Alisson's kick outfoxing Cancelo and putting him through on goal for a second time.
City struggled to break down Liverpool as they tried to fight back in the closing stages, with tensions boiling over when Klopp was red carded by Taylor.
But it was Liverpool who created the better chances, with substitute Darwin Nunez inexplicably opting to shoot rather than find a team-mate during a three-on-one attack, and Alexander Arnold, another substitute, also going close.
In the end, though, a single goal was enough. Enough to kickstart Liverpool's season, perhaps, and certainly enough to leave City with catching up to do on surprise leaders Arsenal.
Source: Skysports