Luis Suarez’s header rebounded off Luke Shaw’s arm to give Barca the lead after just 10 minutes, and the Spanish side never looked back from there.
The visitors dominated possession, despite an impressive showing from Fred in the centre of midfield.
The Brazilian helped United on the counter-attack, and was pivotal in breaking up Barcelona attacks.
Suarez was a constant nuisance for United’s defenders, and was strong for Barca on a night where Lionel Messi was quiet.
Romelu Lukaku spoke to reporters after the game, and fired a warning to his team-mates.
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“Barca are not PSG, they have more experience,’ Lukaku told COPE.
“We have to be confident and go to the Camp Nou with passion and generate more chances.
“We lacked confidence with the ball, I hope that next week we will play better than we did today.”
FRED OUTSTANDING: MAN UTD PLAYER RATINGS
MESSI STRUGGLES: BARCELONA PLAYER RATINGS
Over in Holland, Cristiano Ronaldo fired in his 125th Champions League goal with a bullet header just before half-time.
Juventus’ lead did not last long, however, as Ajax pulled back an equaliser just moments after the restart.
David Neres cut inside from the left wing and curled in a beautiful shot into the opposite top corner.
But Ajax boss Erik ten Hag was still not confident ahead of the return to the Allianz Stadium next week.
“1-1 is not the result you hope for, but it is a result,” he admitted.
“Last time we lost 2-1 to Madrid at home, so we are still in it. We had some very good opportunities.”
RONALDO SHINES: JUVENTUS PLAYER RATINGS
Champions League team of Wednesday night
GK: David de Gea – Manchester United
RB: Nelson Semedo – Barcelona
CB: Chris Smalling – Manchester United
CB: Gerard Pique – Barcelona
LB: Jordi Alba – Barcelona
CM: Frenkie De Jong – Ajax
CM: Fred – Manchester United
RM: Hakim Ziyech – Ajax
LM: David Neres – Ajax
ST: Luis Suarez – Barcelona
ST: Cristiano Ronaldo – Juventus