'The keeper tried to smash him' - Burnley duo left frustrated at Elland Road controversy following Leeds United defeat
Patrick Bamford's fourth minute penalty was enough to secure victory but the visitors were left aggrieved by another first half incident.
Following the opener - where goalkeeper Nick Pope was adjudged to have taken out the Leeds striker - Burnley hit back just before the 20 minute mark but the strike was ruled out with play already halted.
Illan Meslier came to the edge of his box to claim a free-kick from the halfway line but collided with visiting defender Mee in the air.
Referee Robert Jones blew the whistle as Barnes turned to fire home the loose ball with the Clarets protests falling on deaf ears over the award of a free-kick for the hosts.
Bielsa's men then held firm after the break as Sean Dyche's side wasted a number of chances to level.
The Burnley boss confronted the match officials on the pitch post-match, venting his frustration over a number of decisions he felt went against his outfit.
"We had a few good chances, a few decisions went against us, we'll have to fight again," Barnes reflected post-match.
Asked about the incident that saw play stopped before his effort, he said: "The keeper tried to smash Ben Mee, missed the ball and I put it in. I don't know why it was ruled out. There were so many decisions all day, it's frustrating.
"In the second half we were a much better team. We have got plenty more games this season."
Burnley skipper Mee echoed his teammates thoughts.
"I don't know how it's a foul against me, " Mee said.
"If anything, don't give anything - it's more of a penalty than a free-kick. Barnsey put in a good finish, but I don't know what he [the referee] was doing."
He added: "Frustration, decisions didn't go our way today. We started slow, didn't have the best first half but stuck in there and came out to give a good performance."