The Egyptian held off Joao Cancelo, jinked past Bernardo Silva and Aymeric Laporte before firing the ball past Ederson with his weaker right foot.
That put Liverpool 2-1 up but Kevin De Bruyne's curling effort, which took a nick off Joel Matip, ensured both sides shared the spoils.
Earlier, Phil Foden's arrowed effort cancelled out Sadio Mane's second-half opener.
Salah's exploits left Darren Bent spellbound and the talkSPORT host insisted the Reds must get their star man tied down to a big, new contract.
Bent said: "That goal is extraordinary. If Lionel Messi or Cristiano Ronaldo score that, we're talking about it for weeks and weeks.
"He's definitely one of the top two best players in the world right now.
"I'd give him whatever he wants for three years… you have to give him what he wants, he is so important.
"I'd give him £500,000-a-week for three years."
Salah's strike was a goal worthy of winning any match, but Pep Guardiola's side would've felt robbed if they came away empty-handed.
City dominated the first half with Phil Foden forcing Alisson into action on two occasions, while De Bruyne's free header sailed wide of goal.
Meanwhile, Foden was left incensed as his penalty claim was waved away by referee Paul Tierney.
The City man fell in the box after being clipped by James Milner, however, the foul started outside the box and it's not clear if it carried on inside.
Milner arguably got more luck in the second half as his dangling leg took out Silva. Tierney decided not to give him a second yellow card despite animated protests from Guardiola.
Liverpool were a changed team after the break and took the lead just shy of the hour as Mane slotted the ball home - the goal coming thanks to Salah's brilliant run down the right and perfect lay-off to the Senegalese.
It was third time lucky for Foden as he arrowed the ball into the corner of the net after being picked out by Gabriel Jesus.
Liverpool re-took the lead on 76 minutes with Salah's moment of magic, but De Bruyne got City level again.
Both sides could've won it at the end. Rodri made an excellent last-ditch challenge on Fabinho with an open goal gaping, while Jesus' goalbound effort was blocked by Andy Robertson.