It had looked like Mohamed Salah might have produced a Liverpool debut to remember as the winger scored and won a vital penalty for the Reds as they twice came from behind, only to be ousted by a late leveller from the Watford defender.
The Hornets started the contest impressively and the home fans were given something to cheer about early on when Stefano Okaka combined well with Roberto Pereyra, before the latter saw his shot tipped around the post by Simon Mignolet.
Liverpool's defensive organisation proved to be costly for the visitors soon after, as Okaka was left unmarked to break free and head home beyond Mignolet from a curling Jose Holebas corner.
Marco Silva's side played with commendable organisation in the opening half and it took a moment of brilliance for the Reds to find a leveller, as Emre Can produced a standout flick to feed the run of Sadio Mane, who made no mistake with a bending finish beyond a stranded Heurelho Gomes.
The visitors' coordination was to cause more problems just three minutes after though when Watford broke back into the lead, this time as more calamitous defending saw Tom Cleverley cross for Abdoulaye Doucoure to lash home from inside the box.
Watford's full-backs provided a perfect lesson in man-marking for much of the first half, although Holebas was almost made to pay when Salah stole possession down the right, only for Liverpool's new signing to poke over the crossbar with just Gomes to beat.
The Egyptian winger soon made amends for his error after the break by winning a smart penalty off Gomes, leaving Roberto Firmino to equalise from the spot having sent the Watford stopper the wrong way.
It was soon Watford's turn to fall asleep at the back and this time Liverpool were able to exploit the error as Britos played Firmino on side. The Brazilian forward then produced an adroit chip over the head of Gomes, allowing Salah to prod home into an open goal and hand Klopp's men the lead.
The goal clearly gave the Reds a new sense of confidence and they might have added a fourth not too long after, Joel Matip climbing above his marker at a corner before sending a header against the Watford crossbar.
With Watford tiring in the closing stages, Mane sought to take advantage of the hosts' defence when he sprang a counter that allowed Georginio Wijnaldum to take aim from the edge of the box, although he could only skew his shot wide.
It had looked as though Liverpool would squeeze out all three points from the contest, but for Britos who bundled home from Richarlison's knock-down four minutes into injury time in order to hand Watford a remarkable point in Hertfordshire.