Everton suffered a second successive defeat at Turf Moor despite Cenk Tosun’s first goal for the Club.
The Turkey international confidently headed the Blues in front on 20 minutes after the fit-again Seamus Coleman flicked on a Theo Walcott cross.
However, second-half strikes from Ashley Barnes and substitute Chris Wood gave Burnley all three points in freezing Lancashire.
Stand-in skipper Ashley Williams was shown a straight red card late on following a penalty-box tussle with goalscorer Barnes.
Manager Sam Allardyce made three changes to the Everton side that started at Watford the previous weekend.