During a first half dominated by the hosts, Hazard snuck an effort under Gerhard Tremmel from the edge of the penalty area in the 29th minute.
Chelsea failed to build on their superiority, though, and were made to work hard for victory after the break.
Jose Mourinho's men successfully held on to maintain their record of having dropped just two points at Stamford Bridge this season.
Mourinho made five changes to the side that drew 0-0 at Arsenal on Monday, with David Luiz, Ashley Cole and Juan Mata among those coming into the starting XI and Frank Lampard dropping to the bench
Swansea also made a raft of alterations from their 2-1 defeat to Everton, including the introductions of Neil Taylor, Jordi Amat and Angel Rangel - returning from suspension - to the back four in place of Ben Davies, Chico Flores and Dwight Tiendalli.
Taylor had to be alert early on as Oscar found a gap in the Swansea defence, but a poor first touch allowed the Welsh full-back to steal in and make a clearance.
Two minutes later, Branislav Ivanovic had a shot blocked by Ashley Williams, before Jonjo Shelvey dragged the visitors’ first effort wide of the left-hand post from distance.
Chelsea began to turn the screw with a series of corners, and a mis-hit John Terry volley went perilously close to sneaking under Tremmel’s crossbar in the 20th minute. The German did enough to tip the ball over, before Williams was forced to clear from barely a yard out moments later.
The hosts deservedly went ahead just before the half-hour mark, Hazard cutting in from the left wing to send a low effort from just inside the penalty area under Tremmel, who will feel he should have dealt with the strike.
Alvaro Vazquez found himself unmarked at the far post when meeting a flicked Shelvey header minutes later, but his side-footed effort was palmed over by Petr Cech, before Tremmel made an instinctive save to deny Samuel Eto’o and keep the score at 1-0 at the break.
Tremmel was called into action in the opening minute of the second half, making an impressive save from point-blank range to keep out another Eto'o strike.
Swansea looked brighter after that chance, but were unable to cause Cech too many problems.
With Chelsea's dominance waning as the half progressed, Swansea pushed higher up in search of an equaliser, but the hosts soaked up the pressure without ever looking in much danger of conceding.
Indeed, it was Chelsea who eventually had the best chance of the half, substitute Lampard denied by a diving Tremmel save.
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