Goals from Dejan Lovren and Jordan Henderson were enough to give Jurgen Klopp’s side all three points, despite Diego Costa pulling one back in the second half.
Chelsea, who won maximum points from their first three Premier League games, have stuttered in their last two, picking up just one point.
After winning the league comfortably in 2014-15, Chelsea surprised football fans all over the world last season, picking up only 50 points.
Speaking after the game, Conte said:
‘I don’t know if they are still affected by last season but, I repeat, when you finish the season in 10th place, something strange is there.
‘It’s not natural. It’s not simple. When you finish in 10th place, it means that you have to work very hard to improve to make sure you don’t repeat the bad season. We must feel the danger in every single moment of the game.
‘I think that this season, the manager, the players, all the people who work for Chelsea must take their responsibility.
‘For this reason we must pay attention in every single moment of the game, in every minute. Because this is not the first time this has happened.’
Chelsea fell behind in the 17th minute at Stamford Bridge, after Lovren was left unmarked at the back post and allowed to slot the ball past Thibaut Courtois.
Liverpool then doubled their lead through a Henderson screamer from 33 yards in the 36th minute, leaving Courtois with no chance.
Costa gave Chelsea some hope in the 61st minute after prodding the ball past Simon Mignolet and avoiding a desperate James Milner on the line.
However, defeat for Chelsea leaves them in third place on 10 points, having played one extra game and Conte will be hoping Chelsea can prove they have moved on from last season’s failings.
The Blues’ opponents don’t get any easier as their next league fixture is against Arsenal at the Emirates on September 24.