The Reds took the lead through Salah before Jarrod Bowen bagged an equaliser just before half-time. Liverpool came out of the traps well in the second half however, securing the win thanks to goals from Nunez and Jota.
The victory sees Liverpool remain within two points of league leaders Manchester City.
How the game unfolded
It was the visitors who'd create the first real clear cut chance of the game in the sixth minute. Tomas Soucek was found by a Lucas Paqueta cross after ghosting into the middle of the penalty box completely unmarked but, despite finding the corner which his subsequent headed effort, Alisson Becker was able to dive down to his to brilliantly tip the ball wide of the post.
Moments later, Liverpool were caught napping once against by a cross into the box. This time Soucek turned provider for an unmarked Michail Antonio, but his header was a tame one and ballooned well wide of the post.
West Ham were made to rue those chances just 10 minutes later when the referee pointed to the penalty spot after Neyef Aguard clumsily chopped Mohamed Salah down in the box. The resulting spot kick was obviously scored by Salah, who rifled the ball into the middle of the net to make it 1-0.
After taking the lead Liverpool began to control proceedings without really creating many chances. And despite making 100 more passes and having 62% possession, it was West Ham who created the next real opportunity on goal - an opportunity that Jarrod Bowen would take.
Once again a cross into the box caused the Reds problems, as Bowen stole a march on Virgil van Dijk from a Vladimir Coufal cross and canon a fantastic diving header into the bottom corner of the net.
On the stroke of half-time, Liverpool thought they'd re-taken the lead when Dominik Szoboszlai dinked a delightful cross into the back post area. Curtis Jones was on hand to volley it home, but was deemed to have just been offside when the ball was played into the box.
Liverpool came out of the traps quickly at the start of the second half, with Darwin Nunez spurning a glorious chance to give his side the lead in the 52nd minute. The Uruguayan was cleverly found in the heart of the penalty box by Salah, but despite being completely unmarked and having just the goalkeeper to beat he sliced his effort well wide of the post.
Nunez would make amends on the hour mark however. The new Liverpool number nine volleyed home the ball after latching on to an inch-perfect chip over the top of the West Ham defence executed by Alexis Mac Alliser.
The Red were able to firmly secure all three points in the 85th minute when substitute Diogo Jota popped up to tap home a Van Dijk flick down from a corner kick.
Liverpool player ratings (4-3-3)
GK: Alisson Becker - 7/10 - Made a stunning save in the opening five minutes.
RB: Joe Gomez - 5/10 - Caught out by an alarming number of crosses which came into the box.
CB: Joel Matip - 5/10 - Fell asleep a few times when West Ham fired the ball into the box.
CB: Virgil van Dijk - 6/10 - Beaten by Bowen for the equaliser. Bagged an assist for Liverpool's third though.
LB: Andrew Robertson - 6/10 - Ran up and down the left flank in the way only he can do.
CM: Dominik Szoboszlai - 7/10 - He's so good, isn't he? Szoboszlai once again didn't put a foot wrong in the middle of the park.
CM: Alexis Mac Allister - 8/10 - Playing deeper than he was perhaps expected to early in his Liverpool career, but he still found a way to pop up with a brilliant assist for the winning goal on Sunday.
CM: Curtis Jones - 6/10 - An industrious performance from the Scouse midfielder.
RW: Mohamed Salah - 8/10 - Won a penalty and then thumped it home.
ST: Darwin Nunez - 7/10 - Missed a brilliant chance and then quickly made amends by scoring a brilliant goal. His runs in behind were an issue for West Ham on the day.
LW: Luis Diaz - 6/10 - The quietest of Liverpool's front three but still showed flashes of what he's capable of.
Substitutes
SUB: Ryan Gravenberch (77' for Jones) - 6/10
SUB: Cody Gakpo (82' for Nunez) - N/A
SUB: Diogo Jota (82' for Diaz) - 7/10 - Came on and scored again. It's hard to imagine that he'll be on the bench next week.
SUB: Wataru Endo (88' for Mac Allister) - N/A
Subs not used: Kelleher (GK), Tsimikas, Quansah, Konate, Elliott
Manager
Jurgen Klopp - 7/10 - The Liverpool manager has seemingly worked out his early season midfield conundrum, and his team are really firing on all cylinders right now as a result.
West Ham player ratings (4-2-3-1)
GK: Alphonse Areola - 5/10 - Could do nothing about any of the goals, and wasn't really tested aside from the goals.
RB: Vladimir Coufal - 7/10 - Lovely cross to assist Bowen's equalising goal.
CB: Kurt Zouma - 5/10 - Struggled to track Nunez's movement.
CB: Nayef Aguerd - 4/10 - Silly challenge to give away the penalty kick.
LB: Emerson Palmieri - 5/10 - He was up against Salah on Sunday. He tried, but it's nearly impossible to contain the Egyptian star.
CM: Tomas Soucek - 6/10 - Saw a header well saved by Alisson early in the game. Caused Liverpool a few problems with his forays into the final third.
CM: Edson Alvarez - 6/10 - A typically busy showing the middle of West Ham's midfield.
RM: Jarrod Bowen - 7/10 - Bagged his fourth goal of the season with a stunning diving header.
AM: James Ward-Prowse - 6/10 - His delivery into box was impeccable as ever.
LM: Lucas Paqueta - 5/10 - He didn't do too much on the ball during the game, but worked hard off of it at least.
ST: Michail Antonio - 4/10 - Ran around but didn't do a lot bar miss a good chance in the first half.
Substitutes
SUB: Mohammed Kudus (74' for Antonio) - 5/10
SUB: Pablo Fornals (74' for Soucek) - 5/10
SUB: Danny Ings (86' for Paqueta) - N/A
Subs not used: Fabianski (GK), Kehrer, Mavropanos, Ogbonna, Benrahma, Cornet
Manager
David Moyes - 5/10 - West Ham have been very impressive this season and did manage to cause Liverpool problems on the day. Defensive issues, surprisingly, were their undoing on the day.
