Lukaku felt the heat from some Everton fans after his mistake led to Danny Ings cancelling out Eto’o’s early opener in the 20th minute at Turf Moor.
The Belgium striker has not yet had the impact this campaign that he did during his loan spell with the Toffees last season but nine minutes after his blunder, he netted a fourth of the season.
That would probably have been enough as Burnley once again fell short in their search for a first win of the season but Eto’o made sure with a fine third in the 85th minute.
Everton boss Roberto Martinez decided against giving Ross Barkley a third start in a week on his return from injury, with Eto’o starting in his place.
Seamus Coleman, James McCarthy, Leon Osman and Steven Naismith, none of whom started the goalless Europa League draw in Lille on Thursday, were all back in.
Burnley would have hoped to capitalise on any weariness Everton were feeling but instead it was the visitors who took the lead inside four minutes.
Patient passing was rewarded when Osman fed Leighton Baines down the left and the full-back picked out Eto’o, whose powerful header beat Clarets goalkeeper Tom Heaton and went in off the crossbar.
Burnley had played very well in the first half against West Ham last weekend before going down 3-1 but it was a different story this time as they struggled to keep the ball.
Lukaku benefited from such generosity to fire in his first shot of the afternoon, but it slid just wide of the far post.
Gradually the home side began to generate a bit of momentum, with Stephen Ward and David Jones both seeing shots blocked.
But it was Lukaku’s error that allowed them to pull level, inexplicably passing the ball straight to Lukas Jutkiewicz, who timed his through-ball to Ings perfectly.
The striker showed excellent composure to round Tim Howard and slot into an empty net, becoming the first forward to score for Burnley this season.
That lifted the hosts and Scott Arfield was a touch unfortunate to see his well-struck volley fly the wrong side of the post.
However, nine minutes after conceding, Everton went back in front as Lukaku made amends for his earlier error.
The striker received the ball from Naismith just inside the box and turned Jason Shackell. Lukaku’s first effort was blocked by Ward but the ball rebounded to him and this time he managed to guide a shot into the bottom corner.
It did not look the cleanest strike but Everton did not care, and they might have gone into half-time with a two-goal cushion.
Heaton looked to have Naismith’s header covered but flapped at it and was grateful to be able to grab the ball when it came back off the crossbar.
It was a quieter start to the second half but Everton were mostly in control, with Eto’o seeing two long-range efforts comfortably saved by Heaton, who then scrambled Baines’ free-kick round the post.
Burnley boss Sean Dyche sent on Ashley Barnes for Jutkiewicz, while Barkley made his entrance in place of Lukaku with 25 minutes left.
But the spark the home side needed to get back into the game remained elusive and the points were wrapped up five minutes from time when Eto’o turned 25 yards out and curled a terrific shot into the corner.
The Cameroonian almost ended the game with the match ball after seizing on a misplaced back pass and rounding Heaton but his shot hit the inside of the far post and rolled along the goal line.