The hosts dominated possession for much of the encounter, before Juan Cala snatched a precious three points for the Welsh club with a left-footed drive in the 65th minute and David Marshall safeguarded the advantage with two impressive late saves.
Manager Solskjaer feels his charges did enough to merit the win - which leaves them just three points adrift of safety - and declared his relief at registering his first away victory in the Cardiff hot seat.
"It's great," he told Sky Sports. "I've not won a game away from home. It's my first time travelling home with the fans celebrating a win. It's fantastic.
"We did earn it. We did deserve it. They had their chances but we've got one of the best keepers in the league. It was a great effort."
Solskjaer also reserved special praise for his goalscorer, and backed his charges to pull off a "miracle" survival.
"It was a great goal. He's a good footballer," the Norwegian added. "We played with great confidence. I think we showed everyone we believe in this miracle we've spoken about."
