Manager Antonio Conte said Chelsea were “putting pressure” on their rivals as they boosted their hopes of qualifying for next season’s Champions League with victory over Liverpool.
Olivier Giroud expertly headed in Victor Moses’ cross in the first half after the visitors had dominated possession and failed to capitalise.
Chelsea remain in fifth but move within three points of third-placed Liverpool.
“We are winning and our opponents are dropping points,” said Conte.
With two games remaining to Liverpool’s one, Chelsea can move level on points with the Champions League finalists when they face Huddersfield on Wednesday.
But, given their superior goal difference, the Reds would almost certainly ensure a top-four finish with victory at home to Brighton on the final day.
“We had the will to win, to keep the hope alive that we can get a place in the Champions League,” Conte added.
“Probably one month ago it was impossible to imagine we would arrive at the end of the season fighting for the Champions League.
“I don’t know if this will be enough but we have to try. It’s important to face this situation with a lot of pride and show we are still alive.”
Liverpool had more than 66% possession in the first half and created plenty of chances – the best falling to Sadio Mane, who was denied twice by Thibaut Courtois – but it was Chelsea who came closest.
Tiemoue Bakayoko’s header bounced inches wide just moments before Giroud leapt high to nod home Moses’ cross.
And Cesc Fabregas’ smart run in behind the defence created an opening which he could not convert from a tight angle.
Chelsea had the best chances of the second half too. Hazard almost danced his way through before Antonio Rudiger’s header was ruled out for offside and Marcos Alonso’s volley flew inches wide.