Twice Jurgen Klopp's team took the lead through Diogo Jota, and twice the inspired visitors restored parity through Neco Williams and Morgan Gibbs-White in an extraordinary second half.
However, the Tricky Trees could not respond a third time, as Salah moved level with Robbie Fowler in the Reds' all-time scoring charts with the game's fifth and final goal to render Forest's praiseworthy efforts inconsequential.
Readying his side for 90 minutes of Liverpool attacks, Steve Cooper made the bold decision to drop top scorer Brennan Johnson from the first XI in favour of an extra midfielder in Orel Mangala, and the opening exchanges followed the script to a tee.
However, barring an Ibrahima Konate header which went just wide of the post in the 10th minute, Liverpool did little to worry Keylor Navas, who was receiving staunch protection from the 10-strong yellow wall in front of him.
With the Reds' front three found wanting, it was the head of another centre-back which provided the biggest threat, as Navas was forced into action to tip Virgil van Dijk's effort over the bar in the 27th minute from a Trent Alexander-Arnold free kick.
An enforced defensive change would have been the last thing that Cooper wanted as his side frustrated Liverpool, but an apparent rib issue spelled the end of Scott McKenna's afternoon just before the half-hour mark - Joe Worrall replaced his stricken teammate.
The break in play immediately benefitted Liverpool, whose former right-back Williams cleared a Cody Gakpo volley off the line straight after McKenna's withdrawal, and another pinpoint Alexander-Arnold delivery in the 39th minute was wasted by Jota - the Portuguese inexplicably headed wide from a few yards out at the back post.
However, Jota would make amends almost immediately after the restart, and it was another Alexander-Arnold set piece that did the damage, as the Englishman's corner was headed back across goal by Fabinho for Jota to nod home from close range.
The opener was undoubtedly coming, but Liverpool went to sleep after breaking the deadlock and were indebted to Alisson Becker for thwarting Gibbs-White's speculative effort two minutes later, but there was nothing that the Brazilian could do for Forest's swift equaliser.
During a quick foray forward, Gibbs-White laid the ball off to ex-Liverpool man Williams, whose first-time effort took a significant deflection off Andrew Robertson and flew over a sprawling Alisson.
Williams elected not to celebrate, but the Anfield faithful were in a joyous mood in the 55th minute as Liverpool immediately restored their slender lead thanks to some abysmal Forest defending.
An outswinging free kick from Robertson fell kindly for Jota, who drifted away from the Tricky Trees backline unchallenged, chested the ball down and volleyed into the far corner.
In complete contrast to a monotonous first half, a frenetic second 45 showed no signs of slowing down, and Liverpool were caught cold again in the 67th minute, as the Reds failed to deal with a long throw and Gibbs-White's sweetly-struck acrobatic volley found its way into the corner.
Once again, though, Forest failed to shut up shop after levelling the scores, and Salah toe-poked Alexander-Arnold's 70th-minute free kick into the net via the post to bring up goal number 183 for Liverpool.
Neither side were covering themselves in glory defensively during a back-and-forth thriller - especially from long throws in the hosts' case - as Taiwo Awoniyi and Johnson, the latter of whom hit the crossbar, both came close to pegging Liverpool back approaching the final 10 minutes.
Klopp's men ultimately survived to send Forest home empty-handed, and the Reds' second Premier League win on the bounce has propelled them up to seventh place in the table above Brighton & Hove Albion.
Meanwhile, Forest have dropped to 19th place thanks to Leicester City's win over Wolverhampton Wanderers, leaving Cooper's men one point adrift of safety with six games remaining.
Liverpool now have three full days of recovery before a trip to West Ham United on Wednesday, while Forest host Brighton on the same evening.