Luke Shaw’s first-half own goal was all that separated the two sides as the LaLiga giants ran out 1-0 winners in a scrappy game at Old Trafford.
It means the Red Devils will have to beat Barca at the Nou Camp to stand any chance of making it to the last four.
However, despite the enormity of the task that awaits them next Tuesday, United will be aware that they overturned a 2-0 deficit in the last round to complete a famous win over Paris Saint-Germain.
United were outclassed at home in the first leg but managed to make it through in dramatic circumstances but Solskjaer admitted the match in Spain would be even more difficult as he views Barca as the favourites to win the competition.
Speaking after the game, Solskjaer said: “Of course the PSG performance gives us hope and belief that we can do it.
“But we know we are playing probably the favourites to win the tournament and going to the Nou Camp is going to be a challenge.