Jozy Altidore's first Premier League goal for Sunderland came just 14 minutes in to give them the lead, but Chelsea were level moments later when Frank Lampard headed home Hazard's chipped cross in from close range.
Hazard was a thorn in Sunderland's side from the first whistle and it was little surprise when he fired them ahead ten minutes before the break as he cut in from the left before firing beyond Vito Mannone.
Sunderland battled hard and levelled just after the interval when captain John O'Shea fired home as Chelsea failed to clear a corner.
Hazard then restored the lead and it was another wonderful strike as he exchanged passes with Lampard before firing home on the hour.
The game looked to have been won when Phil Bardsley had a lapse in concentration and slid Demba Ba's cross into his own goal with six minutes left.
The Scotland defender made amends and set up a grandstand finish though as he converted from a corner just two minutes later, as Chelsea again failed to defend a simple set-piece.
With six minutes of added time announced the home crowd roared in the hope of an equaliser, but it did not arrive as Chelsea remain in second with Sunderland propping up the rest of the division.
The Black Cats had slipped back to the foot of the table without kicking a ball on Tuesday evening as a result of Crystal Palace's 1-0 win over West Ham, and they emerged determined to do all they could to address that situation in the most trying of circumstances.
Their initial efforts could not be faulted as they took the game to the Blues with Emanuele Giaccherini forcing a first-minute block from Cesar Azpilicueta and Fabio Borini sending a cross across the face of goal seconds later.
But keeper Petr Cech was eventually beaten with 14 minutes gone when, after Andrea Dossena's tame free-kick had come back off the wall, Jack Colback slipped the ball to Altidore and he span John Terry before blasting a left-foot shot past the helpless Cech.
The lead, however, lasted just three minutes when, after Sunderland failed to deal fully with Juan Mata's corner, Hazard recycled the ball, wrong-footed Bardsley and crossed for Lampard to head home unopposed.
Sunderland refused to be cowed by Jose Mourinho's men despite their freedom of movement in midfielder and, with Altidore ruffling the feathers of Terry and Gary Cahill, and Dossena providing a useful outlet down the left, pressed once again.
Cech had to turn away Dossena's stinging 31st-minute effort, which may have been going wide anyway, but opposite number Mannone was not so fortunate six minutes later.
Fernando Torres helped on a long clearance to Hazard on the left and he cut inside past Bardsley and the covering Craig Gardner before firing across the Italian and into the far corner.
But Sunderland refused to give up and they were back on terms within five minutes of the restart when Giaccherini's skidding corner was allowed to reach O'Shea, who swept it past Cech to level.
Chelsea should have been back in front seven minutes later when Ramires robbed Colback and fed Torres, who in turn picked out Hazard in space.
He slid past O'Shea and Ki Sung-yueng with ease before unleashing a shot which Mannone managed to beat away to Torres, who skied his follow-up high over.
But it was Hazard who did the trick once again with 62 minutes gone, collecting Lampard's intelligent back-heel and rounding O'Shea before beating the helpless Mannone with ease.
Bardsley turned substitute Ba's cross into his own net six minutes from time, but then converted Sebastian Larsson's corner at the other end seconds later, although ultimately to no avail.