The first came from youngster James Vaughan, scoring within seconds of coming on as a substitute.
Vaughan has suffered terrible injury problems over the last three years, and this was his first goal for two years.
Marouane Fellaini crossed from the left and the ball eventually fell to Vaughan, who fired in from six yards at the far post on 83 minutes.
The second came from Steven Pienaar when Ayegbeni Yakubu put through the on rushing midfielder, who netted from 10 yards on 90.
Burnley had defender Stephen Jordan sent off with half an hour left, and that decision swung the game Everton’s way and left the visitors still without an away league win this season and deep in relegation trouble.