The Red Devils had taken a 1-0 lead through Juan Mata early on at Old Trafford, but De Gea spilled a simple shot from Antonio Rudiger two minutes before half-time to allow Marcos Alonso a close-range chance to equalise.
But Solskjaer defended the 28-year-old after the match, telling Sky Sports: "David has been unbelievable for this club. We do support each other. There is no chance anyone can blame him for losing points.
"He knows he could have had that shot but that is football. David is one who likes to play games and I will have chats with him and he will respond in the right way."
Asked if he should have dropped De Gea for the game, Solskjaer said to BBC Sport: "No. His reaction after City was good and I was confident in him.
"We did the same against City and started well but today we never found our rhythm again. We needed the tempo to be higher, with a higher intensity but we never managed to get that."
The result leaves the Reds in sixth place – trailing fourth-placed Chelsea by three points and fifth-placed Arsenal by one – with two games still to play.
But Solskjaer is still hopeful of securing Champions League football for next term: "You never know, we felt we needed to win this game. The boys started fantastic, created chances, got in behind and got the lead and then, second half, we never managed to get the quality and tempo back up again.
"It was a good game of football, both could have won it but probably a draw is a fair result.
"They're probably more happy than we are with the draw. It's one of those days and we must dust ourselves down and win the next two – you never know as points are being dropped left, right and centre."