However, the home side will be left to rue missed chances after Kevin De Bruyne failed to convert a penalty in the first-half.
A moment of poor decision-making from Kyle Walker saw Liverpool awarded the early penalty -- the 40th one to be awarded in the Premier League this season -- on 11 minutes after he stuck a leg across Sadio Mane who fell to the ground.
Salah stepped up to take the spot-kick and while he failed to send Ederson the wrong way, he did guide the penalty into the far left corner and out of the City keeper's reach.
Jesus gave City a deserved equaliser on 31 minutes. He received the ball with his back to the goal and turned around Joel Matip before hitting the ball past Alisson.
Referee Craig Pawson awarded a second penalty on 40 minutes when VAR replays showed that a De Bruyne cross had hit the outstretched arm of Joe Gomez.
Despite sending Alisson the wrong way, however, De Bruyne dragged the penalty wide of the goal and the game remained level going into the break.
The end-to-end nature of the game continued with the restart and in-form Diogo Jota forced Ederson into a save on 50 minutes after the keeper initially failed to clear Salah's shot.
It was Liverpool's turn to call for a handball on the hour mark when Jota's shot appeared to hit the arm of Joao Cancelo. However, his arm was judged to have been closer to his body than that of Gomez's in the first-half and no penalty was awarded.
Liverpool captain Jordan Henderson said after the game that he didn't feel like the rules were clear for players.
"To be fair to the referees, the game before [Leicester vs. Wolves] they have been consistent with it," he added. "They are going to the letter of the law. For players it is harsh, I feel for defenders running back for the pace that Kevin De Bruyne cross, it is hard for him to get out of the way.
"Everyone is finding it difficult to know where to put their arms hopefully the refs can look at it and it shouldn't be given in both games."
Trent Alexander-Arnold was forced off moments later after he pulled up with a calf problem while chasing Sterling and Liverpool manager Jurgen Klopp confirmed after the game he would miss England's upcoming Nations League games.
"He will be out of England. Scan tomorrow and then we will see," he said.
The game slowed as the rain continued to fall in Manchester and neither side managed to find a winner.
The result means Liverpool miss an opportunity to go top of the table while City stay in the bottom half in 11th position heading into the international break.
"It was an interesting game, a lot of tactical stuff on the pitch," Klopp added after the game.
"Both teams played on an incredible energy level to close the other team down and use the few opportunities. I thought it was a top, top, top game.
"We were really good and dangerous. We were ready to make little passes. There were moments we played too many long balls. Time to time we just wanted to get rid of the ball."