Hazard's goal was his eighth of the season, seven of which have come in Chelsea's opening eight Premier League games.
Mahrez - who joined City from Leicester for a club-record £60m last summer - wasted a golden chance to win the game after blazing an 85th-minute penalty over the crossbar as the visitors' wait for a league win at Anfield extended to over 15 years.
It is Liverpool's second successive match against a probable title rival after last weekend's draw with Chelsea at Stamford Bridge, which lived up to its billing.
"But that depends on the Liverpool people, not on Manchester City".
Mohamed Salah, who failed to score in a league match at Anfield for only the second time since December 26, then fired over the crossbar at the other end nine minutes later.
Liverpool FC looked to be on course of a valuable point in Naples in the Champions League on Wednesday night before Lorenzo Insigne netted a 90th-minute victor to seal a 1-0 win for Napoli. One situation where you allow them a one-two and then are alone in front of the goalie, or are alone in the box and shoot the ball across the line and somebody scores an easy goal at the back post. We have beaten a good team today.
Asked if the 27-year-old was available, Guardiola said: "We will see tomorrow".
Upfront, the form of Mohamed Salah is a major concern for Klopp, who addressed the slump that the Egyptian talisman is going through right now, insisting there is nothing to worry about for Liverpool fans. We had our chance and we tried to be ourselves'. It is about being consistent every three days during 11 months.
The win was huge for so many reasons, but Gerrard has shown his perfectionist side by emphasising the need to pick up the performances away from Govan.
The loss leaves the Cottagers on the edge of the relegation zone in 17th place, while Arsenal moved up to third above Tottenham on goals scored.
"The challenge for me and the team now is that can we make these home performances away from home and that's when we will find consistency". Liverpool have started so well, they have the same points us, but we were expecting that all along.
"Obviously we won the last three games against City".
"We had our chances in the last minutes to win the game, but we missed it and next time we are going to try to do better".
"You have to be ready, you need a fantastic football team - which thank God I have - with outstanding character, ready to be really fearless, ready to make mistakes. But that's no problem; if you work hard, you earn it". It will always be better to win at Anfield, but do not lose.
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