Chelsea will face a sterner test of their title credentials in Sunday's home clash with in-form Liverpool, while Swansea face fellow strugglers Aston Villa on Saturday.
Jose Mourinho's team, which included five changes from the 0-0 draw with Arsenal, started the better and Hazard played a through ball so delightful that even recipient Oscar was unable to control it and Branislav Ivanovic's deflected shot went over off Ashley Williams soon afterwards.
After Jonjo Shelvey shot wide across goal for Swansea, the lively Hazard did similar after receiving a return pass from Samuel Eto'o and then Ramires fired at Gerhard Tremmel from 25 yards.
Chelsea continued to push forward. Oscar dived in to tackle Alejandro Pozuelo and crossed for John Terry, who volleyed into the floor, with Tremmel tipping over.
It was one of a series of Blues set-pieces and when Swansea failed to adequately clear one cross from Juan Mata, who was back in the starting XI along with Ashley Cole, Terry clipped the ball into the six-yard box and Williams cleared ahead of the lurking John Obi Mikel, whose goalscoring record is less than prolific.
Hazard is Chelsea's top scorer in the Premier League this season and netted his seventh after being granted space to run. The Belgian cut in from the left and hit a swerving right-footed shot, which flew by Williams and Tremmel.
The Swansea captain blocked the goalkeeper's view but Tremmel should have done better.
Chelsea launched an attack as soon as the match restarted after half-time and Eto'o, again with one of several wasted chances, should have scored.
David Luiz's long ball down the right found Mata, who crossed for Eto'o, but from six yards out the striker was denied when he shot straight at Tremmel.
Hazard then found Eto'o on the edge of the area and he shot wide. Mourinho called on reinforcements as Frank Lampard replaced Oscar and Andre Schurrle came on for Mata.
Tremmel turned the ball round the post from Lampard, who shot from his customary position on the edge of the area as again Chelsea were unable to double their lead.
A frustrated Mourinho continued pacing in his technical area and Swansea stretched their hosts on a couple of occasions, without causing Chelsea goalkeeper Petr Cech any alarm.
Hazard went off for Willian for the closing stages and still Chelsea could not find a second but there was no late Swansea flurry, much to Eto'o and Mourinho's relief, as Chelsea triumphed.