Solskjaer masterminded a historic comeback victory away to Paris Saint-Germain on Wednesday night to reach the Champions League quarter-finals.
Marcus Rashford's stoppage-time penalty, awarded with the assistance of VAR, allowed United to sneak through on away goals after drawing 3-3 on aggregate.
Solskjaer has transformed the club's fortunes and though he remains caretaker manager for the time being, he will surely now be named the permanent successor to Jose Mourinho.
"I don’t think I need to say much after all these results, it looks certain," Shaw said after the final whistle.
The story of United’s win in Paris
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"We all love Ole and we love the job he’s done so far. We’re enjoying everything that’s going on at the moment and the results have shown that.
"We’ve had a couple of blips but that’s nine away matches (won in a row) now. They’re not easy games, but it shows what sort of thing Ole and his staff have done. Long may that continue."
United travelled to Paris with a 2-0 first-leg home deficit to overturn and 10 first-team players missing, but Shaw said that he always had hope of achieving an historic comeback.
"I said last week I believed in the team, even with the injuries we’ve had. I felt we were capable of pulling something off and even earlier today I said to Lee Grant that we were going to win, I could feel it.
Shaw believes it 'looks certain' that Solskjaer will be named permanent United manager (PA)
"But that sums up what the mood is like in the camp now and that’s a big plus for the boss and what he’s done in this period of time and how much he’s changed things around. We’ve got so much more to look forward to.
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