Two goals from Messi and one from Phillippe Coutinho ensured the Spanish side ran out 3-0 winners in the second leg of the Champions League at the Nou Camp.
Barca advanced to the semi-finals of the competition on a 4-0 aggregate win having won the first leg 1-0.
Speaking after the game, the United boss heaped praises on Messi for his spectacular showing in the clash.
“I have to say Lionel Messi is top quality and he was the difference of course,” Solskjaer told BT Sport. “At 2-0 straight after it was game over.
“He’s different class, he and Cristiano Ronaldo are the best players of the last decade, everyone agrees on that one. Messi showed his quality.”
He added: “We have to aspire to get to that level of Barcelona, we can get there but we have loads of work to do.
“If we want to get back to Man Utd’s true level, true traditions, we have to challenge Barcelona.
“They were a couple of levels above over the two games.”