Barkley appeared to raise a hand in the direction of an opponent, prompting the referee to issue a straight red card.
Everton coach Alan Stubbs told evertontv: "Ross's football was outstanding at times today but he made some silly decisions after some really good pieces of play and was trying to do too much when he could have played it simple.
"He's let himself down with the sending off. He shouldn't have given the referee the dilemma as to whether to send him off.
"He hasn't punched the boy but he has raised his arm, which you can't do. I can see why [the referee] has sent him off, so I've got no complaints."