Stoke goalkeeper Lee Grant had made a series of fine saves from a rampant home side before Cesc Fabregas's pinpoint flighted corner found Cahill's head at the near post and the captain nodded home.
But barely a minute into the second half Bruno Martins Indi poked in an equaliser after Peter Crouch headed down a free kick by Charlie Adam.
Willian restored the lead twice in a frantic seven minutes either side of a 64th-minute strike from Crouch before top scorer Costa capped new year celebrations for Antonio Conte's side with Chelsea's fourth goal.
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(Reporting by Clare Lovell, editing by Ed Osmond)
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