United created some decent chances in the first few minutes, but didn't take them, and were punished when Ashley Young gifted Lionel Messi possession - who proceeded to score a brilliant goal from outside the box.
His first goal was had plenty of the familiar Messi magic, as he ghosted past a defender and bent the ball just inside the upright.
The second was largely thanks to a blunder by David de Gea.
"It was an important goal for him after the criticism he has received".
Barca sit nine points clear at the top of La Liga with six games remaining and will play a Champions League semi-final against Liverpool or Porto before taking on Valencia in the Copa del Rey final on May 24.
"Messi and Cristiano [Ronaldo] have been the best players in the world for the past decade", said Solskjaer.
"And he showed why we think that and why he has won so many titles".
United were threatening at times over the two legs but it was clear to see there was a big difference in quality between the two teams.
"Now we'll go step by step, little by little, focus on LaLiga Santander, we won't relax, and then we'll approach the semi-finals with the focus is demands".
"We are happy, but we don't talk about achieving the treble", said the Frenchman.
"Cristiano increased our chances of winning the Champions League but football isn't math", Allegri added.