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Alvaro Morata struck either side of an Andros Townsend equaliser, with Eden Hazard providing the assist for the second on his return to the side.
Pedro converted from Marcos Alonso’s cross with 20 minutes left to seal the win.
However, Sarri says his side can still improve and thinks they still have work to do to reach the levels of their title rivals.
“From the first day, Manchester City and Liverpool are at the moment a step ahead of us,” he said.
“When I arrive the gap from last season is 30 points with City.
“It is difficult to think you can recover this gap in one season, in six months.
“We need to improve more because we are doing well.
“We need to improve from a tactical point of view.
“In the first 15 minutes of the second half we were not compact.
“In the defensive phase we need to improve.
“When we can’t cover the ball in the opponent’s half we are in trouble. So we need to improve.”
Sari was pleased to see Hazard make his comeback after missing the last three games through injury.
He took just one minute to lay on Morata’s second of the afternoon.
Morata now has four goals in his last four Premier League games – as many as he managed in all of his previous 23 appearances in the competition.
And Sarri also reserved special praise for the Spanish striker.
“I am really very happy for Alvaro,” he said.
“He improved in the last month in confidence and personality.
“He improved also from a tactical and technical point of view. He plays with more continuity from the team.
“He has great potential physically and technically. He can improve more. Potentially he can be (a top striker at a top club).
“Sometimes he is a little bit fragile.
“We were a little bit in trouble so we needed some quality and personalty.
“We were lucky because we had Eden [Hazard] on the bench. He was able to immediately change the match.”
Morata should have had a hat-trick but fluffed his lines by chipping straight into the hands of Wayne Hennessey when through on goal right at the death.
And he admitted that the miss was playing on his mind despite being pleased with a brace.
“I can score a hat-trick so it was not the perfect day but we will take the three points,” Morata said.
“That is the most important thing for us. We want to fight until the end for the Premier League.
“We played good today. I think a lot about the hat-trick but I am improving and I want to keep helping my team to win.”