Luis Diaz opened the scoring in the first half and Salah netted with a penalty after the interval at the Amex Stadium.
Klopp’s side have won their last eight league games, keeping the pressure on City, with the champions playing at Crystal Palace on Monday.
Salah has reached 20 goals in a Premier League season for the fourth time, with his latest strike coming at just the right time amid his tense contract negotiations.
Klopp said on Friday it is up to Salah whether he signs a new deal, a claim that was followed by the Egypt forward’s agent tweeting a series of laughing emojis just minutes later.
Salah’s long-term value to Liverpool is immense, but of more immediate concern to Klopp will be the knock that saw him replaced by Diogo Jota soon after his first goal in four games.
With a testing trip to Arsenal looming on Wednesday followed by an FA Cup quarter-final at Nottingham Forst, Klopp will hope his star makes a quick recovery.
“Something was not right. I think there was a situation where his foot got stretched. We’ll have to see,” Klopp said.
While City are in fine form, Liverpool’s hot streak is equally impressive.
The Reds have scored 21 goals and conceded just twice in their winning run, while also keeping a third successive league clean sheet for the first time this term.
When Salah converted from the spot, it made Liverpool only the second club to score 2,000 goals in the Premier League, emulating Manchester United.
“It was expected to be a difficult game and I’m really happy with certain moments. We were really strong and scored a great goal,” Klopp said.
“We had massive counter-attacks that we didn’t use but it’s fine. I’m really not overly critical with the boys.
“Consistency is the most important thing, but it’s also the most difficult as well.”
On the title race, Klopp added: “It’s always the same, you expect Manchester City to win every game when you watch them play.
“The only thing we can do is as win as many football games as possible. We’re ready, we don’t need to call it a title race, we just want to win games.”
Liverpool, chasing an unprecedented quadruple, suffered only their third defeat this season against Inter Milan in the Champions League last-16 second leg in midweek.
But Klopp insisted his players are such bad losers that they were certain to mount an immediate response to the Inter loss, which didn’t stop them progressing to the quarter-finals on aggregate in any case.
– Incisive Salah –
The German’s assessment proved correct and they took the lead in the 19th minute.
Diaz sprinted onto Joel Matip’s perfectly-weighted pass and bravely headed into the empty net as Sanchez rushed off his line, pole-axing the Colombian with a dangerous high challenge in the process.
Diaz was unable to celebrate as he was writhing on the turf after being caught in the chest by Sanchez’s leg, but the keeper surprisingly escaped a red card following a VAR check.
Diaz’s second goal since signing from Porto in January was almost followed by one for Salah, whose incisive burst ended with a low drive that forced Sanchez to save at his near post.
Leandro Trossard shot wastefully over from Neal Maupay’s defence-splitting pass soon after the interval.
Salah almost made him pay for that miss with a deflected effort that looped onto the crossbar.
Liverpool’s luck was out on that occasion but they doubled their lead in the 61st minute.
Naby Keita’s shot was blocked by Yves Bissouma’s arm, conceding a penalty that Salah drove past Sanchez with ease to keep the title race bubbling nicely.