Jamie Vardy put the Foxes ahead inside just three minutes - only the fourth goal the Reds have conceded in 11 Premier League matches at Anfield this season - as they endured a sloppy start.
But Liverpool dominated the remainder of the first half with goalscorer Salah guilty of wasting a couple of fine opportunities inside the box while Leicester goalkeeper Kasper Schmeichel thwarted Philippe Coutinho and Roberto Firmino.
Yet the hosts kept plugging away and finally drew level through Salah on 52 minutes, collecting Sadio Mané's back-heel and beating two defenders to fire low through Schmeichel.
And Salah repeated the trick on 76 minutes when he received James Milner's pass and rolled Harry Maguire to go one-on-one and slide beyond the 'keeper, sealing Liverpool's first come-from-behind win this season.
German hails Liverpool's reaction and defensive display
"You don't want to be 1-0 down but if you are 1-0 down that early in the game you want to see the reaction I saw tonight," Klopp told reporters.
"It was the best reaction we showed so far after being 1-0 down. It was fantastic, I didn't see one nervous player, I didn't see any real reaction in a negative way. We kept on going and created and created and created [chances], and played brilliant football.
"We were defensively really strong. The main strength of Leicester is counter-attacking and quick transitions and if you are 1-0 down it plays into their hands and makes it more difficult. But we did really well and I liked the first half a lot apart from the result, to be honest."
Klopp even felt that Liverpool's performance was one of their best of the season, having shown grit and desire to claim an important win.
"Everybody in the stadium reacted in the right way, everybody saw, 'wow, what a game!' It was 100 per-cent one of the best we have played so far," he continued.
"We knew at half-time we had to carry on, there were not a lot of judgments that we had to change this or that. We [said we] have to do it again and again and again, and keep our nerves, that's the most important thing.
"Of course you get a little bit annoyed after missing the third or fourth chance but they will come again. As long as we don't concede, we always have the chance to score.
"That's what we did; fantastic goals, two fantastic goals. They felt so good and felt so fair in this moment and deserved. So I’m really happy about it."
Klopp: Van Dijk addition has not made players nervous about their places
Joël Matip's error led to Vardy's early opener with Virgil van Dijk, who will officially join Liverpool on Monday in a £75 million deal from Southampton, watching on from the stands.
Yet Klopp insists his current defenders should not be worried about their places in his squad, insisting that Van Dijk's arrival is designed to help ensure Liverpool are prepared for injuries or losses of form.
Asked whether he would need to reassure his defenders that they still have a part to play after the Dutchman's arrival, the Reds boss declared: "Look, that's part of professional football.
"The really good players will always say that the challenge with other players for a position in the team will always help you. That's how it is. We cannot go in a season with two centre-halves and just hope nothing happens.
"The third one is already logical and if you have a fourth one then it's even better. No player should and can play a whole season. At a club like Liverpool, we need to react in situations when a few of them are maybe not [at their] top level or injured. That's what we did.
"I don't think the boys were nervous, I couldn't see anything of that tonight. They don't have any reason to be nervous because I am really happy about the development of these players.
"But we have to obviously fight and battle with the biggest teams in world football, so we cannot go there with a group of 11 [players] being best friends every day. We need to have a squad which is ready for that. That's what we are building step-by-step."