Haaland increased his haul to 32 goals to tie Salah's mark for a 38-game season - and the Norwegian scoring machine has eight games still to play.
Only four days earlier, Haaland had become the first player in English football to score a record 45 goals in a season in all competitions during City's 3-0 win over Bayern Munich in the first leg of their Champions League quarter-final.
On Saturday, that record tally increased to 47 goals.
Pep Guardiola (53) sat out his star striker for the second half at Etihad Stadium on Saturday, giving him some rest before the second leg of their Champions League showdown with Bayern.
Guardiola's side, who are unbeaten in 14 matches in all competitions, trail leaders Arsenal by three points in the race for the Premier League title that appears to be heading for a pulsating conclusion. Arsenal play West Ham on Sunday.
Guardiola was also quick to heap praise on his star striker after the game.
"I want, I desire, he can break all the records as possible," Guardiola told a news conference. "That means he's scored a lot of goals, that means that helps us.
"But I think he wants to win the titles, to be there, and still we are there.
"But it's impressive, (we) still have to play eight games and he's close to breaking all of them."
Kalvin Phillips (27) added to the BBC: "I think we are very happy with three more points. Three more points closer to Arsenal and we are chasing them.
"Our standards slipped a little towards the end there were a few mistakes and we had a couple of substitutions. Overall we are happy with the first 60 minutes or so."
John Stones (28) put City ahead in the in the fifth minute. With the ball flying around the box like pinball, Stones got his left foot on it to beat keeper Daniel Iversen (25).
Haaland made it 2-0 with a penalty in the 12th minute, awarded due to Wilfred Ndidi's handball. He calmly slotted it into the bottom right corner.
He scored his second in the 25th minute after Kevin De Bruyne (31) steamrolled past Ndidi to feed Haaland, who finished with a chip past Iversen.
Trailing 3-0, former City man Kelechi Iheanacho (26) pulled one back for Leicester in the 75th minute on otherwise nightmare afternoon for their interim manager Dean Smith, whose team are 19th with 25 points from 31 games in their battle to avoid relegation. They are two points adrift of the safety zone.
"There's lots we can take," Smith told Sky Sports. "I'm obviously disappointed but a two-goal deficit to the best team in the world isn't the end of the world.
"Our season won't be defined here, we have a run of games where we have to win matches."
Leicester had a couple more late scoring chances. City keeper Ederson stopped James Maddison's point-blank shot after a one-on-one. Iheanacho's long shot in injury time clanged off the post.
