Argentine Carlos Tevez denied the visitors what would have been a memorable victory, slotting home from the penalty spot on the stroke of half-time.
"It was a sensational performance, the boys were absolutely terrific," Round told evertontv. "I think to be the third game in a week like that and to put the effort in and the commitment that they did, it was truly Everton today. I was really proud of them.
"I thought we dominated the game in the first half and got a deserved lead and then second half they showed what they're all about to be fair. They've got great quality and fantastic players.
"You have to play against what they put out against you and the boys did that. They stood up to it and kept going. It was the effort and commitment, coupled with the courage to play in that arena under that pressure that I was really pleased with.
"We actually looked against Arsenal and here like we're a top four team."