The west Londoners took the lead through the Belgian, though struggled to extend their scoreline against a stubborn Swans back line, which rarely threatened the home side.
During a first half dominated by the hosts, the Belgium international darted infield before unleashing a powerful drive that deceived goalkeeper Gerhard Tremmel in the 29th minute.
Chelsea failed to build on its superiority, though, and was made to work hard for victory after the break, though it held on successfully to maintain its record of having dropped just two points at Stamford Bridge this season.
Jose Mourinho made five changes to the side that drew 0-0 at Arsenal on Monday night, 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 its 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 as Oscar found a gap in the Swansea defense, but a poor first touch allowed the Welsh fullback 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 poorly 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 was unable to cause Cech too many problems. With Chelsea's dominance waning as the half progressed, Swansea pushed higher up in search of an equalizer, but the hosts soaked up the pressure without ever looking in much danger of conceding.
Indeed, it was Chelsea which eventually had the best chance of the half, with substitute Lampard denied by a diving Tremmel save.