United were second best throughout the second leg of their Champions League quarter-final clash as Barca ran out 3-0 winners to ease through the tie 4-0 on aggregate.
Messi fired home from range after just 16 minutes and doubled the lead thanks to a dreadful error by United keeper David de Gea.
Philippe Coutinho confirmed the inevitable when he curled in on the hour mark and Solskjaer admits his side are nowhere near where they want to be after watching his side fall to a fourth consecutive away defeat in all competitions.
“I have to say Lionel Messi is top quality and he was the difference of course,” the Norwegian said. “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.
“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.”
United started brightly and nearly took an early lead when Marcus Rashford crashed an effort on to the crossbar after just 40 seconds.
However, Barca never really relinquished control and nearly punished United for spurning their early chance on 11 minutes as they were awarded a penalty that was overturned by VAR.
They did not have to wait long for the opening goal, though, as Messi skipped inside from the right before smashing into the bottom corner from just outside the box.
Their lead was doubled four minutes later when Messi took aim from range once more.
This time he opted to go with his weaker right foot and his tame effort should have been easily gathered by De Gea.
Instead, the Spaniard let the shot creep under his body and the tie was effectively over as any sort of contest.
With Barca in cruise control, the biggest surprise was that it took them so long to make it 3-0.
When they did it was worth the wait as former Liverpool star Coutinho curled into the top corer to cap a superb night for the visitors and a miserable one for Solskjaer and his struggling side.
United can now concentrate on trying to secure a top-four finish in the Premier League as they look to secure qualification for next season’s Champions League.
They face Everton away on Saturday before two tough home games, with Manchester City to visit Old Trafford next Wednesday and Chelsea four days later.
“They were a couple of levels above over the two games,” Solskjaer added.
“We want to play these games again next year so we have a massive week coming up. There is no point dwelling on what could have been, we have to focus on that week.”
