Mane 's 29th-minute effort had cancelled out Stefano Okaka's eighth-minute header, only for Abdoulaye Doucoure to immediately restore the home side's lead.
Miguel Britos added to Liverpool manager Jurgen Klopp's frustrations with a late equaliser that ensured Watford salvaged a point following a 3-3 draw at Vicarage Road on Saturday.
Klopp had spent the previous 24 hours handling the fall-out from Philippe Coutinho's transfer request after the club rejected Barcelona's £90 million ($117.1 million, 99 million euros) bid for the Brazil playmaker.Liverpool remains adamant Coutinho will not leave and it looked like Klopp might find relief from the saga when debutant Mohamed Salah gave the visitors the lead after they had twice come from behind. Sadio Mane 's 29th-minute effort had cancelled out Stefano Okaka's eighth-minute header, only for Abdoulaye Doucoure to immediately restore the home side's lead.
Liverpool, who were without Coutinho, were back level in the 55th minute through Roberto Firmino's penalty.But with Salah's 57th-minute goal looking as though it would be decisive, Britos bundled the ball home deep in stoppage time after Simon Mignolet had pushed Richardson's shot against the bar.
For all the close-season transfer dealings, both starting lineups carried a familiar look, with Nathaniel Chalobah the home side's only new face and Salah the lone Liverpool debutant.
For Klopp's side, it appeared to be a case of same old faces, same old problems, as they allowed Watford to twice take the initiative during the first half on the back of defensive errors.
The opening goal was certainly one to frustrate Klopp as Okaka took full advantage of Liverpool's lack of organisation as they failed to defend Jose Holebas's corner.
Roberto Pereyra had broken threateningly after being sent clear by Okaka and was unfortunate to see his shot deflected wide off Dejan Lovren following a twisting run that had opened up the visitors' defence.
Holebas swung in the resulting corner towards the near post, where it was headed home by Okaka, who had been allowed to advance unchecked, with Roberto Firmino in particular guilty of failing to react.It was the perfect start for Watford manager Marco Silva, but for Klopp, it was an early reminder of the problems that need to be eradicated if Liverpool is to prove genuine title contenders.
The other side of Liverpool was demonstrated 21 minutes later when Mane levelled in spectacular fashion.The Senegalese forward, who started the game on the left, picked up the ball on the right and worked his way across the pitch before exchanging passes with Alberto Moreno.Moreno played the ball into Emre Can, whose flick set up Mane for a precise finish.
It was an incisive breakthrough, but the good work was almost immediately undone when Watford made their way too easily down the right-hand flank in the build-up to Doucoure's goal.Doucoure was at the heart of the move, releasing Tom Cleverley, whose low cross was met with an attempted clearance by Trent Alexander-Arnold.
But the ball ricocheted off team-mate Joel Matip's back, allowing Doucoure to add the home side's second.Once again Liverpool had been responsible for their own plight, although Watford's positive approach was unsettling Klopp's side.
The visitors' response in the second half confirmed the attacking strengths of this Liverpool side.Within 10 minutes of the restart, the Reds were back level after a superb flick by Firmino sent Salah clear.The winger was brought down by Heurelho Gomes as he attempted to round the goalkeeper and Firmino finished from the penalty spot.Three minutes later, Firmino again produced a moment of exceptional skill as he made up for his error on the opening goal.
The striker brought down Moreno's floated pass before lifting the ball over the advancing Gomes and Salah ensured the chance was not wasted by bundling home from close range. Liverpool's victory looked assured, only for Britos to reward Watford's persistence at the death.