De Gea allowed a tame right-footed shot from Messi to squeeze under his body in the 20th minute to make it 2-0 to Barcelona on the night and 3-0 on aggregate.
The goal left Manchester United with a mountain to climb at the Nou Camp at the half-time break.
And Solskjaer was spotted by BT Sport cameras heading back to his seat in the dugout amid an air of resignation.
De Gea’s shocking blunder came less than five minutes after a bit of magic from Messi fired Barcelona into the lead on the night.
The Argentina superstar nutmegged Fred on the edge of the box before pushing the ball into space and firing past De Gea in the 16th minute.
Barcelona had a penalty ruled out by VAR before Messi’s brace.
Referee Felix Brych pointed to the spot after Fred appeared to bring down Ivan Rakitic in the box.
However, Brych overruled his initial call after consulting VAR.
In other Man Utd news:
Barcelona earned a 1-0 win at Old Trafford in the first leg of their last-eight tie last week.
BT Sport pundit Steve McManaman believes De Gea took his eye off the ball.
McManaman said: “What can you say about that one? It says it all really.
“It was a backpass from Lionel Messi that found its way into the back of the net.
“The goalkeeper takes his eye off it, it’s a lack of concentration, and the ball agonisingly rolls into the back of the net.
“Left-foot, right-foot, he just needs a header now.
“Manchester United have contributed to both goals they conceded early on.
“I said earlier it needed to be the perfect performance and it’s been anything but. Now they have to be careful they don’t go too gung-ho and leave themselves too often at the back.”
Former Premier League midfielder Robbie Savage believes De Gea’s blunder spells the end of the road for United in this season’s competition.
“Oh no no no. David de Gea,” he said on BBC Radio 5 Live. “It was such a weak effort from Lionel Messi and it just trickles under De Gea. Game over.”