The Red's have struggled to get going this season but tonights opponents helped Liverpool go back to their old ways as Klopp's men dominated from the first whistle.
Trent Alexander-Arnold opened the scoring in the 7th minute through a majestic free-kick into the top corner.
Allan McGregor's heroics kept the scoreline at a respectable level through a string of superb saves, denying Nunez, Jota, Salah and co on eight occasions.
Salah then doubled the hosts lead from the penalty spot, slotting home to give Liverpool back to back wins in the Champions League. The Merseysider's now sit on 6 points, 3 behind group leaders Napoli.
Here are how the players rated:
Jurgen Klopp may have a lot of issues to sort out but his goalkeeper is certainly not one of them. Was largely a bystander for most of the game but when called upon late on, he expertly denied Antonio Colak from close range to ensure a much needed clean sheet.
Silenced his critics through a superb free-kick early on. His passing range caused problems all night for Rangers and was a much improved performance defensively. Form is temporary but class is permanent.
Virgil Van Dijk- 7
A vast improvement from Brighton. Looked comfortable and had confidence in abundance, winning 100% of his Aerial duels (6). Calm as you like.
Joel Matip- 7
Like his partner, Matip won 100% of his Aerial duels (4). Was rarely troubled and gave Alfredo Morelos zero joy.
Always a threat down the left but his final ball was lacking. An incredible last ditch clearance of the line in the dying minutes was lauded by the home crowd and helped protect the clean sheet
A much improved performance from the skipper, setting the tempo from the off. Was full of energy and was more creative playing a deeper role, threading two pinpoint passes to carve open the Rangers backline.
Kept Liverpool on the front foot and controlled the tempo along with Henderson. The two of them helped keep the midfield compact and looked more defensively solid. The Spaniard won 5/6 of his ground duels as well as having a 100% dribble success rate and winning 100% of his Aerial duels.
Luis Diaz- 8
His change of pace and trickery left the Rangers defence helpless. His final ball could have improved however, striking it straight into the palms of Alan McGregor in the first half. His explosiveness led to him being clattered in the box, winning a penalty for his side.
Given plenty of space on the right which he would have found a pleasant suprise. Went close in the first half with a curling effort which was tipped behind by McGregor. He got on the scoresheet in the second half through a composed penalty down the middle which made him the joint 2nd top scorer for an English side in Champions League history with 35 goals. He is now level with Thierry Henry despite playing 18 less games.
Darwin Nunez- 8
Looked confident and more threatening in this change of shape. He was denied four times by McGregor but Liverpool fans should be encouraged by his movement and the ability to fashion so many chances. He is a chance magnet and the goals will come.
Diogo Jota- 8
Constantly found pockets of space and combined well with Nunez, he showed his creative side playing four key passes. Had a thunderous strike tipped over the bar.
Roberto Firmino- 6
Helped to keep the tempo high but didn't see a lot of the ball.
Brought on for the captain to help maintain control which he did successfully. He will hope that he did enough to force his away back into the 11 against Arsenal on Sunday.
Harvey Elliott- N/A
More minutes and more experience gained for the youngster.
James Milner- N/A
Subbed on for his 300th appearance for Liverpool. Made a superbly timed challenge on Colak late on which was greeted like a goal from the crowd.
Joe Gomez- N/A
Came on to give Alexander-Arnold a deserved standing ovation.
Allan McGregor- 8
Was comfortably Rangers best player, making eight saves, denying Darwin Nunez four times as well as tipping a Diogo Jota rocket over the bar and clawing Salah's effort wide. Was sent the wrong way for the penalty but if it wasn't for the 40 year old veteran, it could have been 4 or 5.
James Tavernier- 5
When Tavernier struggles, Rangers struggle. And that was the case tonight. The Englishman offered very little going forward and was pinned back by Diaz. Failed to leave his mark on the game.
Looked nervous, needlessly putting the ball out for a corner multiple times. Made a great headed clearance to deny Nunez in the second half.
Ben Davies- 5
The former Liverpool man finally made his first appearance at Anfield but it was not one to remember. He fouled Nunez early on, giving away the free-kick which was scored by Alexander-Arnold.
Leon King- 5
Not an easy game for your first Champions League start. His inexperience was shown, tripping Diaz in the second half giving away a penalty.
Borna Barisic- 4
Struggled against Salah and was not at his usual best. Failed to offer anything going forward.
Did not impress against his boyhood club and was poor in and out of possession, giving the ball away and being overrun. Booked for a cynical foul on Jota.
Steven Davis- 4
Poor in possession and did not have the energy to press Liverpool. Struggled to make any impact.
Ryan Kent- 5
The ex Liverpool man was Rangers most dangerous attacker and superbly rolled Henderson twice. He didn't see too much of the ball however and couldn't really get at Liverpool's defence.
Didn't look comfortable playing out of position on the right and was largely anonymous. Looked out of his depth.
Alfredo Morelos- 3
Did not have a sniff and his threat was completely nullified by Van Dijk and Matip. Failed to even register a shot before he was hooked.
Ryan Jack- 4
Couldn't really make any impact.
Nearly converted his strike at the death but his effort was saved by Alisson. He would have disappointed not to start.
Had his effort cleared off the line heroically by Tsimikas.
Glen Kamara- N/A
Fashion Sakala- 4
Added directness and ran at the Liverpool defence but his strike was poor.