Cuthbert earns superb Chelsea slender advantage on Barca
Chelsea defended brilliantly at the Olympic Stadium to beat a Barcelona side aiming to win the trophy for the third time in four seasons.
Jonatan Giraldez's side, who hammered Chelsea in the 2021 final and shaded a tight battle in last season's semis against the Blues, were left frustrated in their first home defeat in over five years.
Hayes is hoping to crown the end of her time with Chelsea before taking over the US women's national team, with her first Champions League triumph as a coach.
After defeats in the FA Cup and the League Cup final, Chelsea's bid for a quadruple has crumbled in the last few weeks, while Barcelona's is alive but dented by Cuthbert's fine goal.
"We know Barcelona are the best team in Europe and everything has to be perfect for you to get close to winning a game of football," Hayes told reporters.
"We took our chance and did a lot to limit their chances. You can only do that together, not individually, and the team as a whole did that today."
Hayes' gameplan worked to perfection as Chelsea reduced a usually dominant Barcelona to remarkably few chances while threatening occasionally on the counter-attack.
Having never beaten the Catalans before, the Blues set up with three centre-backs and plenty of grit.
They soaked up Barcelona pressure without letting the two-time winners work many dangerous openings.
'Minor details'
Salma Paralluelo had a sight of goal but her shot was deflected over, while Ballon d'Or winner Aitana Bonmati was kept under control.
Chelsea shocked the hosts by capitalising on the first real attacking move they managed to string together five minutes before half-time.
Sjoeke Nusken manoeuvered well in the box, turning and finding Cuthbert, who worked some space before arcing a shot beyond Cata Coll.
Barcelona came back out strongly and thought they had earned a penalty when Kadeisha Buchanan handled Patri Guijarro's shot.
Referee Stephanie Frappart was summoned to watch a VAR replay and controversially determined the offside Paralluelo was interfering with play, just behind Buchanan.
Barcelona, in the semi-finals for six consecutive seasons, summoned two-time Ballon d'Or winner Alexia Putellas from the bench to try and break down Chelsea's defence.
Mayra Ramirez appealed for a penalty at the other end as she went down under a challenge from Irene Paredes and then blazed a fine chance for Chelsea's second over the bar.
Paralluelo volleyed wide with five minutes remaining as Chelsea held firm under increasing pressure.
Putellas should have levelled with virtually the last kick of the match but fired wide from close range with the goal gaping after a free-kick was nodded down to her in the box.
Barcelona fans tried to rouse their team for the second leg at Stamford Bridge on April 27 by chanting "Yes we can" at full-time.
"It's Barcelona, they are the champions -- they will be angry," said Hayes, anticipating a strong response from Giraldez's side.
The Barca coach, also heading to the US at the end of the season to take over Washington Spirit, said his team would keep their nerve.
"We're calm, we still have 90 minutes ahead," said Giraldez.
"A draw would have been perhaps fair -- there were few chances and in football when things are balanced, minor details make a difference."