It was Elliott's goal in May that lifted the Clarets back into English football's top-flight after an absence of 33 years but few Burnley fans can honestly say that they would have expected to have claimed successive wins over two of last season's top five, Manchester United in midweek and now Everton.
The Clarets almost took a first-minute lead when Chris McCann's cross was headed against the crossbar by striker Martin Paterson and goalkeeper Tim Howard did well to save the rebound from Steven Fletcher.
Paterson should also have scored in the third minute but this time glanced his header from Elliott's cross wide.
It took Everton half-an-hour to get into the game but then they created two chances in the space of a couple of minutes.
Leighton Baines' corner was met by the head of Jack Rodwell but his downward header was well saved by Brian Jensen.
Marouane Fellaini then slipped a lovely pass behind Robbie Blake to the overlapping Tony Hibbert but his low cross to the far post just evaded the boot of striker Louis Saha.
The only goal of the game arrived when Everton were enjoying their best spell of the game. Blake's cross from the left found its way to Fletcher who pulled the ball back for Elliott to score with a left-foot shot beyond Howard.
The Everton goalkeeper did well to push an overhead kick from McCann over the crossbar on 62 minutes but the Toffees were presented with a glorious chance to square the game when, 16 minutes from time, they were awarded a penalty by referee Phil Dowd.
He adjudged that Hibbert had been brought down by McCann. It seemed a harsh decision but perhaps justice was done when Saha pulled his kick wide of the target.
The Everton striker almost made amends with 11 minutes remaining but shot wide form the edge of the box.
The result leaves David Moyes' side at the bottom of the Premier League table still seeking their first point of the season.