The Reds' record signing netted shortly before the hour mark with what looked set to be the winning goal, only for Miguel Britos to nod the ball in following a corner after goalkeeper Simon Mignolet failed to parry the ball away from his line.
Stefano Okaka had headed Watford into the lead and Abdoulaye Doucoure restored the home side's advantage shortly after Sadio Mane scored a superb equaliser for Liverpool, who were without injured Barcelona target Philippe Coutinho.
The hosts were good value for their lead in Marco Silva's first match in charge but the erratic Heurelho Gomes brought down Salah early in the second half to allow Roberto Firmino to equalise from the penalty spot.
Salah turned the ball home from close range two minutes later to put the visitors in front for the first time but, just as they looked primed to claim all three points in Jurgen Klopp's 100th competitive match in charge, they were undone by more slack set-piece defending, as Watford grabbed a point.
The Hornets gave the Liverpool defence a scare seven minutes in, with Roberto Pereyra denied by a good block from Dejan Lovren after racing in behind Trent Alexander-Arnold.
The visitors failed to heed the warning, though. From the resulting corner, Okaka headed the ball into the net off Simon Mignolet from inside the six-yard box, with four Liverpool defenders standing and watching the striker.
Silva's side were good value for their lead but Liverpool snatched an equaliser of the highest quality with their first shot of the match. Mane's one-two with Moreno was flicked back into the Senegal forward's path by Emre Can, and he swept the ball high past Gomes from 12 yards out.
Parity lasted less than three minutes, though. Tom Cleverley was played into the area by Okaka and his low cross from the right was kicked against the head of Joel Matip by Alexander-Arnold, allowing Doucoure to blast home past Mignolet.
Salah missed a clear chance for a debut goal in the first half but the Egypt international helped turned the game on its head within the first 12 minutes of the second half.
Gomes raced from his line to bring Salah down inside the penalty area, allowing Firmino to send the Watford keeper the wrong way from 12 yards and make it 2-2 10 minutes after the break.
Barely 90 seconds later, Watford's defence was undone by a simple long ball to Firmino, and, as his lob over the stranded Gomes threatened to bounce wide, Salah arrived to bundle the ball over the line.
A ferocious Alberto Moreno half-volley was tipped just over before Matip rattled the crossbar with a header, as Liverpool suddenly looked ready to run riot against a Watford side who appeared to be tiring.
Gomes kept out a Lovren header and Salah blasted over from 20 yards, but there was a nervy moment for Mignolet – confirmed by Klopp as Liverpool's number one for the season – as he fumbled a high ball before gathering at the second attempt.
Watford applied some pressure, with Mignolet saving well from debutant Richarlison, but the hosts made the most of slack defending once more to salvage a point from the resulting corner.
Liverpool failed to clear their lines from the delivery and Richarlison's shot came off Mignolet's outstretched arm and cannoned off the woodwork, with Britos on hand to head the ball over the line and spark wild celebrations in the stands.