Jurgen Klopp's men were close to securing all three points at Vicarage Road but a late goal from Miguel Britos rescued a point for Marco Silva's men.
Mane, Roberto Firmino and Mohamed Salah grabbed the goals for the Reds but some sloppy defending cost them the victory.
The Senegal international claims his side deserved to win but he has conceded that they will need to improve ahead of their Champions League qualifier against Hoffenheim on Tuesday.
"We're disappointed because we deserved to win," he told Liverpool's official website. "We should have defended better in this kind of game, it's always complicated with long balls. It's the first game of the season and we're going to try to work hard for the next games.
"No excuses -- everybody is fit. We're unlucky we didn't get three points because we were looking for that, but the next game we're going to be ready to get three points to make our fans happy."