Hazard’s effort proved the difference between the teams, but Chelsea’s failure to build on the Belgium international’s ninth goal of the season meant it continued the run of recent underwhelming displays by the Blues.
Chelsea’s performance in grinding out a scoreless draw at Arsenal three days previously was described as a return to an approach that had been familiar during manager Jose Mourinho’s first spell in charge.
Mourinho dismissed suggestions that the performance had been negative, but having consolidated their position within striking distance of the leaders, they needed victory over a stuttering Swansea to build on that hard-earned point.
With the visit of Liverpool looming in three days, Mourinho took the opportunity to make five changes to his starting line-up, including the return of Ashley Cole for his first league start since the defeat at Newcastle United on November 2.
Up front, Samuel Eto’o replaced Fernando Torres in the role of main striker, although the Cameroon international showed few signs of ending Chelsea’s misfortunes in front of goal.
Torres, Eto’o and Demba Ba have scored just five league goals between them so far this season and the Blues were once again forced to rely on their goal threat elsewhere in the team as they struggled to turn their dominance into goals.
Swansea arrived at Stamford Bridge having won just one of their last seven games and rarely looked like ending that frustrating run.
With Michu, their main striker, out for around six weeks after undergoing surgery on an ankle problem, Michael Laudrup’s side offered little going forward.
Instead they were forced onto the back foot for long periods and had to rely on a combination of Chelsea’s poor finishing and increasingly desperate defending to maintain a foothold in the game.
Ashley Williams, the Swansea centre-back, in particular was outstanding, making a succession of timely clearances.
And while goalkeeper Gerhard Tremmel might have done better for Hazard’s goal, he made an eye-catching save to deny Eto’o twice either side of the interval.
The breakthrough came after Chelsea had already applied sustained pressure on Tremmel’s goal.
Branislav Ivanovic saw a shot deflected over the crossbar and Williams’s hooked clearance denied John Mikel Obi the chance to score from close range after John Terry had headed down Juan Mata’s deep cross.
A goal appeared imminent, although Laudrup will be unhappy with the way his side conceded.
Cole sent Hazard clear on the left wing, from where he cut inside, gliding easily past Jordi Amat, before hitting a low right-foot shot from the edge of the area that appeared to go under Tremmel’s arms.
A minute later the goalkeeper was again at fault and fortunate to escape after being hurried into a mistimed clearance under pressure from Eto’o.
Swansea offered little by way of response, although Alvaro Vasquez should have done much better when he found himself unmarked after being picked out by Jonjo Shelvey’s header.
As dominant as Chelsea had been, they needed a second goal to eliminate the chance of Swansea nicking a point from a rare foray forward.
Eto’o was twice offered the chance to double the lead, but was denied by Tremmel on both occasions.
The first time, three minutes before the interval, the goalkeeper saved after Eto’o had been sent clear by Mata.
And Mata was again the provider 12 seconds after the restart when he whipped in a right-wing cross towards his team-mate, whose close-range volley was tipped away by Tremmel.
Chelsea’s nerves began to show as Swansea went in search of an equaliser late on, but they were not forced to pay for their earlier missed chances.