The visitors drew level early in the second half through Nicolas Jackson firing beyond Caoimhin Kelleher after VAR overturned the initial offside decision.
But Curtis Jones restored the Reds' advantage just three minutes later when he met Salah's cross to the far post and guided it past Robert Sanchez.
Arne Slot's side consolidated their one-point cushion as league leaders following Manchester City's late win over Wolves earlier on Sunday.
Here were the key talking points from Anfield:
Until now, Arne Slot had been largely untested as Liverpool head coach.
A shock defeat to Nottingham Forest notwithstanding, the Premier League's fixture calendar proved largely favourable to him ahead of Chelsea's visit.
Contests between the Reds and their London counterparts are often cagey affairs with a series of goalless draws punctuating the previous two seasons.
But with the hosts seeking to reclaim top spot and Enzo Maresca's side winning all of their away fixtures to date, the stakes were considerably higher.
Manchester City's late victory at Wolves threw the pressure back on their Anfield rivals with Arsenal already at risk of falling into the middle distance.
Slot's charges responded to the gauntlet by evoking memories of a not-too-distant past when Rafael Benitez regularly jousted with Jose Mourinho.
They seized key moments to reclaim the lead after Nicolas Jackson's early second-half equaliser and weathered the storm during the closing stages.
Kopites stopped short of chanting '...and now you're gonna believe us' but the top flight should be sitting up and taking note of their team as title contenders.
Curtis Jones definitely won't forget this past week in a hurry.
The Toxteth native had several reasons for sleepless nights ahead of this encounter and not exclusively attributable to his early days of parenthood.
A second Premier League start of the season threatened to be an arduous one up against his one-time England under-21 teammate Cole Palmer.
Lesser figures may have been cowed by the prospect of keeping the league's reigning Young Player of the Year under wraps amid a rich vein of form.
Not only did he limit Palmer's goal threat but also thrived himself by winning one penalty and should have claimed a second before VAR's intervention.
Later, a superb sense of anticipation saw him beat the offside trap to meet Mohamed Salah's equally impressive cross-field pass and net the winner.
Jones has endured his fair share of knocks in a senior career blighted by injury and performance issues but truly seized his opportunity here.
When Liverpool and Chelsea did battle under Benitez and Mourinho in the mid-2000s, they were contests which were underscored by their intensity.
Such was the intensity of the occasion between the clubs that it briefly threatened to become English football's modern equivalent to El Clasico.
The landscape has shifted considerably since then, with City a greater thorn in Anfield's side over the previous decade but those days may soon return.
Needling proved a constant theme when north and south clashed, and this latest meeting had it in abundance with players and staff caught up in it.
In total, referee John Brooks issued eight bookings with even Slot cautioned following a touchline outburst during one of the first half's testier moments.
Darwin Nunez's name also entered the book after he replaced a struggling Diogo Jota and later engaged in a running battle with Benoit Badiashile.
Both clubs may have moved on different trajectories in recent years but this encounter may prove to be the reigniting of their once ferocious rivalry.