Dusan Tadic boosted Southampton's bid to beat the drop as his double sealed a crucial success in the south-coast derby at St Mary's.
Bottom club Albion are five points from safety and an eight-season stay in the top flight would still be over if they fail to take maximum points from their last two games against Tottenham and Crystal Palace.
Fabregas' curling finish was a fitting way to bring up his half-century of goals in the Premier League, but it was Hazard's clinical exploitation of a Swansea error that made it, pouncing on Andy King's sloppy loss of possession, weaving his way forward and picking out the right pass.
How did the Chelsea players perform during their win over Swansea City?
Suddenly, Chelsea looked more vulnerable, but Swansea have struggled to score goals all season.
Swansea will feel they might have had a penalty in the second half when Gary Cahill trod on substitute Nathan Dyer's foot, but referee Jonathan Moss waved away their claims.
Mike van der Hoorn got in the way of a goal-bound Victor Moses strike soon after, while Alfie Mawson clipped his own crossbar attempting to clear a loose ball as Chelsea dominated the opening period.
The result leaves Chelsea FC in fifth place in the table but just two points behind Tottenham and six behind Liverpool FC as the race for Champions League qualification hots up.
Trump Actually Just Said Name-Calling Is a 'Disgrace'
Jackson's nomination had become imperiled even before Capitol Hill Democrats on Wednesday released new allegations of misconduct. Jackson withdrew his name from consideration for the nomination Thursday before his position on Choice could be explored.
Draghi Faced Nowotny Plea to Discuss ECB Policy
Later in the press conference, Draghi said they have an unchanged confidence about reaching the 2 percent inflation target. He also said the ECB's support to the economy was still needed and the central bank would reinvest bonds.