Salah's 76th-minute strike came soon after he missed a one-on-one with City goalkeeper Ederson, but the forward maintained his cool to secure a statement victory for Jurgen Klopp's side -- ending City's run of 14 unbeaten matches this season.
"Our position is not the best and it is just one game so we have to carry on. Hopefully this gives us more confidence," Salah said.
"Today we created chances and when I missed the first I was calm because I knew the second would come. I just tried to stay focused and put it in the net because I missed the first one, and I got lucky."
City started with their usual dominance in possession and forced some early openings which Joe Gomez dealt with adequately.
The highly-anticipated clash exploded into life straight after the restart. First, Salah broke free and raced through on goal, but a great save by Ederson sent the strike trickling wide.
Soon after, Phil Foden looked to have opened the scoring with a strike deflected off Gomez flying into the top corner which Liverpool protested after the goalscorer seemed to have kicked the ball out of Alisson's grasp.
The goal was eventually ruled out but not for the foul Liverpool appealed for -- instead, Anthony Taylor chalked off Foden's goal due to Haaland's infringement on Fabinho in the build-up.
"The referee said play on, play on, play on, there were a thousand, million fouls like this and this one is because we scored a goal," Pep Guardiola said. "So they disallowed because we scored a goal, otherwise it would not have been disallowed.
"We lost because we make a mistake but this is Anfield."
Immediately after the drama, Liverpool came close to opening the scoring themselves as Salah teed up Diogo Jota from just yards out, but the Portuguese forward nodded his free header off target.
Klopp's side posed a bigger threat in the second half and that paid off 14 minutes from time as Salah opened the scoring, sending Anfield into raptures.
Alisson did well to blast the ball forward and isolate Joao Cancelo with the Egyptian forward, the latter won the battle and raced through, this time getting the better of Ederson and stalling City's chase for the Premier League top spot.
The match had time for one more moment of drama, and it came in the form of a sending off as Klopp saw red for his protests against referee Taylor.
Salah was pulled to the ground by Bernardo Silva but Taylor remained undeterred, sending Klopp charging down the touchline to argue against the decision -- Taylor waited for the ball to go out of play before brandishing a red card to the German manager as Liverpool saw out the victory as Darwin Nunez squandered several chances to extend their lead.