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...
5. Charlie Taylor 6
Had his work cut out against Iwobi, who caused problems with his pace when hitting the byline. Wasn't as tight down his side of the pitch as we've come to expect and couldn't dig out a delivery to really trouble Everton's rearguard.
6. Robbie Brady 7
A very encouraging return for the Republic of Ireland international, who makes a real difference when he's on song. Neat and tidy in possession, delivered well into the penalty area, tracked back well and took his goal superbly.
7. Ashley Westwood 7
Kept Burnley's midfield ticking as the hosts got on top prior to Delph's withdrawal. That changed the complexion of the game, but Westwood was still influential in and out of possession. Pressed the ball well and broke up the play to tee up Brady's opener.
8. Josh Brownhill 6
Wasn't too far away from his first Premier League goal for the Clarets when firing just wide of the post in the first half. Telegraphs the play nicely and his ability to turn the ball over in the opposition half is a major plus point for Burnley.