Dominic Calvert-Lewin cancelled out a fine opener from Robbie Brady to share the spoils in this North West derby at Turf Moor.
After just three minutes, Brady, who had been in and out of the team as a result of form and injuries, caught Jordan Pickford out with a low 25-yard strike on his weaker right foot. The ball didn't move at a lightning speed but it was an accurate effort which nestled in the far corner.
Alex Iwobi has been a big benefactor of the wing-back system, finding pockets of space in advanced areas. Everton started in a 3-4-3 shape and it allowed the Nigerian to freely burst forward. His fizzing ball into the box was prodded towards goal from Calvert-Lewin but his international colleague, Nick Pope, made a great reflex save.
The Burnley full-backs were finding it tough to deal with Everton's system in an attacking sense.
Matt Lowton was easily spun by Richarlison but the Brazilian's finish was tame as he broke into the box.
Brady was playing with a lot of confidence and the Irishman did well to work an opportunity for Josh Brownhill. The former Bristol City man dragged his effort wide from just outside the box.
Everton had to switch to a back-four after Fabian Delph pulled his hamstring, midway through the first half, with Andre Gomes coming onto the field. The Spaniard doesn't have the skillset to play as a wing-back and so 3-4-3 shifted to 4-3-3 with Iwobi and Ben Godfrey becoming right-back and left-back respectively. The tactical shift actually helped the visitors get a stronger hold on the midfield.
Richarlison, as ever, was a willing protagonist for Everton. He again powered through the centre of the field but he sliced his shot well wide. A player who is capable of the brilliant and the bizarre.
On 40 minutes, the Everton defence went to sleep after a long punt from Pope. Dwight McNeil reacted quickly to get the ball under control and play Chris Wood through on goal. Wood looked destined to score but Pickford flew off his line to make an excellent sprawling save.
Allan won a midfield tussle with Ashley Westwood before releasing Richarlison in behind. This time, the 23-year-old illustrated outstanding end product, bending a delivery between James Tarkowski and Ben Mee with Calvert-Lewin diverting the ball into the net.
It was almost a carbon copy of Calvert-Lewin's earlier chance. Pope saved the first one but he couldn't do anything about this one. A fox in the box.
Burnley would have been frustrated at the break but they were indebted to a valiant block from Lowton at the start of the second half.
Everton won the ball high up the pitch and Burnley struggled to clear. The ball broke to Gomes in the box but Lowton threw himself across the box to make a vital block.
Burnley were looking to press the visitors but Everton were breaking through that press.
James Rodriguez moved untracked to the edge of the box and, when the ball was eventually worked to him, he forced Pope into a wonderful one-handed save, clawing the ball out of the top corner.
Burnley fought back and Josh Brownhill moved his feet quickly in a crowded penalty area but his strike was deflected behind. The Clarets looked to crowd the penalty area from set-pieces but Everton dealt with them well.
Pope had put in another fine performance as Gareth Southgate watched on from the stands but he was almost caught in no-mans land as Richarlison released Calvert-Lewin. But the goalkeeper used his huge body-frame to get in the way of the attempted finish.
Southgate would also have been fairly impressed with his current choice as England number-one. Although he conceded another goal from distance, Pickford did pull off some good stops and, in the latter stages of the second half, he made a smart diving save to flick Wood's header to safety.
At the end, Gylfi Sigurdsson, a late substitute, looked set to win the match after latching onto a brilliantly intricate through ball from Rodriguez. In his way, almost inevitably, was Pope, who made a brilliant block with his foot.
In the end, both teams will reflect on what might have been but a point was a fair result.
Burnley: Pope 8; Lowton 7, Tarkowski 7, Mee 7, Taylor 5; Brady 7 (Benson 86, n/a), Brownhill 7, Westwood 7, McNeil 6; Rodriguez 5 (Barnes 75, 6), Wood 6.
Everton: Pickford 7; Godfrey 8, Mina 7, Keane 7; Iwobi 6, Allan 7 (Tosun 89, n/a), Doucoure 7 (Sigurdsson 81, n/a), Delph 5 (Gomes 29, 6); Rodriguez 7, Calvert-Lewin 7, Richarlison 8.
Nick Pope: A battle of the two England goalkeepers and Pope came out on top. Pickford had a good game but Pope made more saves.
Ben Godfrey: He has been a brilliant acquisition since arriving from Norwich City, mainly because of how versatile he has been in the middle of this injury crisis.