Klopp's side had surrendered pole position a few hours earlier when Manchester City won 3-1 at Crystal Palace, piling pressure on the Reds to respond as the title race heats up.
But Liverpool rose to the challenge as Sadio Mane's opener five minutes after the break and a blistering long-range drive from Salah two minutes later moved them two points clear of second-placed City on a raucous afternoon at Anfield.
"It was a fantastic goal from Mo. I had the best view," Klopp said after Salah's first goal from outside the area in the league since January 2018.
"What a finish, what a shot. But the whole move around it. Wow, brilliant.
"I'm so proud of the team, it was a fantastic performance. I'm so thankful I can be a part of this. It's overwhelming at times."
City have five games left compared to just four for Liverpool, leaving the destiny of the title in the hands of the champions.
But there is a growing feeling at Anfield that this will be the season Liverpool finally end their 29-year wait to be crowned kings of English football.
With City having difficult league fixtures looming against Tottenham and Manchester United, Klopp's men are convinced they can hold onto pole position.
Their run-in is undoubtedly less daunting than City's, with Cardiff, Huddersfield and Newcastle on the schedule before a potential title party against Wolves at Anfield.
"The first question in the meeting today was 'what is the City score?'. You cannot avoid knowing about it," Klopp said of a gripping title battle.
"We expect them to win all their games so we just need to get as many points as possible.
"If we're champions then great, but if not we are still a really good football team."
For fourth-placed Chelsea, the loss was a blow to their bid for Champions League football next season and they will drop to fifth if Arsenal win at Watford on Monday.
Claiming that Liverpool's first goal should have been ruled out for a foul on Emerson, Chelsea boss Maurizio Sarri grumbled: "In my opinion there was a foul. I think the level in Premier League is the best in the world but not for the referees.
"We stayed in the match for 50 minutes. It's not easy of course but we have to fight to the end (for the top four)."
On the eve of the 30th anniversary of the Hillsborough disaster that killed 96 Liverpool fans and in the week that club legend Tommy Smith passed away, there was even more emotion at Anfield than usual.
After a minute's silence featuring mosaics reading "30 years" and "96" held up by fans around the ground, Liverpool's focus was trained on avenging one of the more painful defeats in the club's recent history.
- Sweet moment for Salah -
Steven Gerrard's infamous slip in 2014 led to a defeat against Chelsea that effectively gifted the title to Manchester City when Liverpool had looked odds-on to lift the trophy.
But Klopp insisted his players wouldn't be haunted by the ghosts of that collapse and they were true to his word.
"We have closed the book on that now, Klopp said about Liverpool's Chelsea demons.
The crucial opener arrived in the 51st minute.
When Salah scampered onto Roberto Firmino's flick in the Chelsea area, Blues defender Emerson could only prod the ball to the influential Jordan Henderson and his deft cross reached the unmarked Mane, who headed home at the far post.
Any Liverpool nerves had been wiped away by jubilant celebrations and soon after Salah blew the roof off Anfield with a moment of pure genius.
Taking possession wide on the right flank, Salah cut inside and glided away from Emerson before unleashing a sumptuous 25-yard drive that arrowed past Kepa Arrizabalaga into the far corner.
It was a sweet moment of vindication for the Egypt forward after a group of Chelsea fans were filmed chanting "Salah is a bomber" before their team's Europa League tie at Slavia Prague on Thursday.
And Salah, a practising Muslim, celebrated his 22nd goal of the season by adopting a prayer position in front of the Kop as Liverpool moved a step closer to their holy grail.