A disappointing afternoon for Everton worsened late on when centre-back Ashley Williams was sent off for elbowing Barnes, the Wales worldwide now facing a suspension with Sam Allardyce's men still having work to do to make absolutely sure of safety.
Cenk Tosun scored his first goal for the club in the first half, but second half goals from Ashley Barnes and Chris Wood helped the Clarets grab their first win in 11 games, causing Everton fans to turn on manager Sam Allardyce.
Allardyce has guided Everton from the brink of the relegation zone to the top 10 in the Premier League table but there has been speculation he could leave the club at the end of the season.
"My job is not what the fans are saying or not saying".
Allardyce prompted more boos from the seething away end when he hooked Sigurdsson in favour of Yannick Bolasie with seven minutes left but the last incident of note wouldn't be an Everton equaliser but the latest chapter in Williams's catalog of shame. "It shouldn't happen, he's swung his arm and he's an experienced player", he said.
"I think it was such a big game today for us to win to go level with Burnley".
"We were on the front foot, particularly from the second half".
Jordan Pickford kept Everton in front with three excellent saves in front of England manager Gareth Southgate but Ashley Barnes finally beat him in the 56th minute and Chris Wood marked his return from injury by heading in the victor 10 minutes from time.
"We need to show some desire first and foremost, we do have some quality players that should shine through that".
"And then we decided not to deal with the corner correctly and allow a free header".
Dyche's Clarets had previously never recovered from a goal down to win a Premier League game during his tenure: per Opta, Burnley had previously trailed during 60 games, drawing 12 and winning none. "We have to say to ourselves, 'why did we lose that?' and it's us, we've only got ourselves to blame".
In the last 53 games in which Burnley fell behind they had failed to win but there was a determination and positive intent to Dyche's side on Saturday, despite a recent lack of victories.
"Burnley have a poor record coming back from behind, but if you have a poor record who do you want to come up against?" That's not an easy thing to do. Seamus Coleman returned at right back after overcoming a thigh problem while Leighton Baines took a place among the substitutes.
Burnley remained a threat themselves as long as they were allowed to get crosses into the box from the flanks and it took a tremendous save from Jordan Pickford to keep the home side out with half an hour gone. It was a thorough performance and winning one. "Hopefully that freedom in the second half is something we can carry on".