South Korea's Ji So-yun scored with a fine free-kick for Chelsea but it was not enough to defeat European powerhouse Lyon who had gone ahead through Eugenie Le Sommer's well-taken goal that secured a 1-1 draw in the second leg and 3-2 aggregate victory.
French side Lyon will now face Barcelona in a bid for a fourth succesive Champions League crown.
Hayes, while proud of how Chelsea had pushed Lyon hard, made no attempt to hide her unhappiness with Persson's performance.
"I'm going to take a fine for it, but I thought the referee was horrendous," said Hayes.
"You really want strong officials in big games. She didn't manage the game, I think she was scared of them. She didn't control both teams."
Chelsea pressed hard for a second goal in the second half with captain Karen Carney curling a fine shot against a post.
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Defeat left the London club unable to qualify for next season's Champions League as Chelsea cannot finish in the top two of this term's English Women's Super League.
"Unfortunately we've been too much of a nearly team this year," admitted Hayes.
"We've gone toe-to-toe with the champions of Europe, we scared the life out of them and made them average," she added.
"Lyon's seven-year dominance is massively threatened. It's the start of a new era in the women's game. We just would have liked to have progressed and put the nail in their coffin today.
"But I can't be critical, I'm so proud of my players."
Lyon boss Reynald Pedros accepted Hayes' assertion that his side's European dominance is now under serious threat.
"Our rivals are getting stronger," he said. "We're very much aware, we knew from our experiences once you get to the knockout stages, this year and last year, that other teams are getting stronger.
"We know we've got to work even harder than ever to stay ahead of the pack."
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