Defender Hayes struck a long-range free-kick that sailed over half of the field and into the net to reduce the second-half deficit, however Fran Alonso's side could not find an equaliser and exit the competition despite late pressure.
Glasgow City, who won their own qualifier at home in Cumbernauld, have been Scotland’s representatives since the UEFA women’s cup was rebranded 12 years ago, joined by Hibs on three occasions. But Celtic’s debut in the competition, via finishing runners-up in last season’s SWPL, ended after just one game. They will face Minsk in the third-placed play-off on Saturday after an encouraging performance.
Alonso’s side were behind at the break when Sandie Rose Toletti tucked in the opener, no less than the Spanish side deserved from the opening half hour. Chloe Logan saved a penalty after half an hour to keep the scores blank but could do nothing about the opener seven minutes later.
Alba Matia Redondo Ferrer added the second shortly after the restart before Hayes answered back with a wind-assisted set-piece that caught everyone out.
A rallying end gave hope for the Scottish side but a late chance for Izzy Atkinson went wide as Scottish participation halved in Trondheim.
It was a dominant first-half display that ensured victory for Glasgow City against Maltese visitors Birkirkara in their first Uefa Women’s Champions League match of the campaign earlier in the day.
Clare Shine got Grant Scott’s Scottish champions off to a flying start at Broadwood Stadium in Cumbernauld with a 30-yard screamer just three minutes into the morning contest.
Costa Rican star Priscilla Chinchilla made it 2-0 on 15 minutes with a curled strike before the edge of the area before Israeli international Vital Kats, signed just two weeks ago, netted her first goal for the club shortly before half-time.
City will be back in action at Broadwood on Saturday evening against BIIK-Shymkent as they look to secure a place in the play-off round to get into the group stages. The Kazakh champions saw off Slovan Bratislava 4-0 in their own game at Clyde’s Cumbernauld home.