The Reds looked set for a fourth successive Premier League victory but Simon Mignolet was too easily beaten by Willian's cross-cum-shot which looped over him and into the far corner just 144 seconds after the Brazilian - who also scored twice against Qarabağ FK on Wednesday - had come on to the pitch.
That cancelled out Salah's earlier close-range side-footed finish, his 10th goal in 13 league games, from Alex Oxlade-Chamberlain's prodded through ball which had put the hosts ahead.
The Egyptian also went close to firing a late winner, and his third successive league brace, only to be denied by a strong Thibaut Courtois save.
Chelsea have now gone without defeat at this ground since May 2012, remaining unbeaten on all seven of their last visits - the last four of which have all ended by a 1-1 score-line.
Liverpool could have moved level on points with the London side with a win, but instead will be rueing yet another match in which their defence succumbed to maintained pressure.
Fast-paced start despite European exertions
Both teams made five changes apiece with Liverpool's selections the more surprising - Sadio Mané and Roberto Firmino both benched in favour of Oxlade-Chamberlain and Daniel Sturridge.
Chelsea brought Álvaro Morata back in after benching him for the 5,000-mile round trip to Baku in the Champions League in mid-week, with captain Gary Cahill and Tiemoue Bakayoko also among those to return to the starting eleven.
Both teams were distracted by European commitments earlier in the week but a fast-paced start belied any fatigue.
The Reds started brightly with Salah consistently involved, an early shot blocked by Chelsea captain Cahill before he fired high and wide from Sturridge's lay-off on the edge of the area.
Fresh from back-to-back 4-0 wins over West Bromwich Albion and Qarabağ, Chelsea soon moved through the gears with the free-flowing Eden Hazard the architect of their attacking promise in the first half.
Drinkwater squanders one-on-one
Danny Drinkwater - on his first league start for Antonio Conte's men - went close to the game's opening goal when released by Hazard.
The Belgian burst forward with typical brilliance and delivered a well-weighted through ball, but the former Leicester City midfielder failed to round Mignolet - who stuck out a leg to deny him.
A goalmouth scramble followed that as Mignolet pushed Davide Zappacosta's powerful shot over before the resulting corner saw the ball ping across the box without ending up in the back of the net.
After Marcos Alonso curled a free-kick wide at the Kop end, Ragnar Klavan had to be alert to clear Zappacosta's first-time cross when Alonso had lifted a ball into the Italian in space on the right side of the area.
Salah created Liverpool's best first-half chance all by himself in a near-identical recreation of his goal against Southampton here, rolling free from Cahill and curling a left-footer towards the back post - though this one bent just wide.
Hayden Christiansen headed wide at a corner after the interval while Joe Gomez was fortunate that his failed clearance from another Zappacosta cross on a dangerous counter-attack put off Bakayoko and Morata, both lurking at the back post.
Morata saw a Drinkwater cross inadvertently come off his chest and go wide of the target courtesy of a couple of deflections from Liverpool centre-back pairing Klavan and Joël Matip.
Salah nets yet again to fire Reds in front
Liverpool then found the breakthrough and yet again through Salah, who netted his 15th goal of the campaign across all competitions in this just his 20th appearance - and against his previous employers.
The Egyptian, who never looked like realising his promise in an underwhelming spell at Stamford Bridge that ended only in the summer of 2016, ran on to Oxlade-Chamberlain's poked pass - following good work from Philippe Coutinho - and slotted low beyond Courtois to make it 1-0 on 65 minutes.
But Chelsea upped the ante after going behind and Conte's changes - introducing Cesc Fàbregas and Pedro - ensured a sustained spell of pressure on the Merseyside outfit.
Another introduction, Georginio Wijnaldum, had a shot deflected wide at the other end but it was the Blues who looked the more likelier to score with Liverpool dropping deeper and inviting pressure.
Klavan timed his offside trap to perfection with Morata looking to put Hazard one-on-one with the goalkeeper after another threatening break as Chelsea searched for their equaliser.
Substitute Willian punishes Liverpool
Gomez got caught beneath Fàbregas' long ball forward, only to recover and tackle well, as Liverpool's defensive weaknesses began to show and the rising tensions on L4 eventually resulted in a leveller.
Willian, introduced just two minutes earlier, strode into the box before looking to stand up a cross towards the back post. Mignolet misjudged the ball's arc and saw it loop over him and into the bottom corner.
Despite the confidence of that goal it was the Reds who almost nicked a late winner, but Courtois was equal to Salah's effort after he controlled James Milner's pass to shoot from 15 yards.
For the second time in a week, Liverpool were unable to protect a lead late on - having thrown away a 3-0 lead at Sevilla, conceding a stoppage-time equaliser - and miss the chance to leapfrog Tottenham Hotspur into fourth.
Chelsea could be 11 points off Manchester City by the end of the weekend and are now three points behind Manchester United, who they beat earlier this month, in second.
Despite failing to hold on to win, conceding just their second goal of the season on home soil, Liverpool remain the only team Conte has faced, and failed to beat, in the Premier League.
There was at least positive news for Jürgen Klopp's side in that midfielder Adam Lallana made his first appearance of 2017-18 off the bench late on after overcoming a long-term thigh injury.