Robbie Brady put the home side's noses in front when beating England international Jordan Pickford from 20 yards out.
Chris Wood saw a huge chance go begging when he failed to beat the Everton stopper from Dwight McNeil's threaded pass.
And then the Toffees drew level on the stroke of half-time when Dominic Calvert-Lewin converted Richarlison's cross.
Both sides had chances to win it - as Nick Pope denied James and Gylfi Sigurdsson while Pickford kept out Wood's header - but the spoils were shared.
Here are the ratings...
9. Dwight McNeil 6
Knitted play together nicely when moving inside and switching the play. Played a perfectly measured pass through for Wood in the first half and looked to drive with the ball a bit more in the second. Will be disappointed to have missed the target with a shot late on.
10. Jay Rodriguez 6
The forward was excellent for half-an-hour in the first half, but struggled to pick up space and provide the link between midfield and attack when Everton changed shape. Replaced by Barnes in the 75th minute.
11. Chris Wood 5
Missed a huge opportunity to double Burnley's lead before the break, a miss that would be punished by the visitors. Affected the game when pinning Godfrey, but was no match physically for Keane and Mina. Went close with a header late on.
12. Ashley Barnes 5
Was only given 15 minutes to affect the game, but put himself about and made a nuisance. Used his strength to hold the ball up and develop openings in the final third.