Gregor Townsend's much-changed side were in danger of a shock defeat after falling 17-7 behind at the interval, but Scotland stormed back in the second half to avoid a major embarrassment at Murrayfield, clinching a bonus-point win.
Dobie scored two tries after the break to spark Scotland's revival, while fellow replacements Pierre Schoeman and Scott Cummings also crossed over.
The Scots' escape act secured their second win from three matches in the new tournament. They defeated Argentina 47-38 in their opener before losing 42-28 in South Africa last weekend.
Scotland will resume their Nations Championship campaign in November with matches against New Zealand, Australia and Japan.
Townsend's decision to make 14 changes looked like coming back to bite him as Fiji dominated the first half.
Jonny Gray's close-range try and George Horne's conversion gave Scotland an early lead, but Fiji were unfazed.
Tevita Ikanivere touched down following a maul to cap an immediate response from Fiji, who struck again through Selestino Ravutaumada's sensational effort.
Taking possession wide on the flank, Ravutaumada kicked ahead and recovered the ball before racing away to score.
Elia Canakaivata pounced on the loose ball after a break down and jumped over the line for another Fiji try, with Isaiah Armstrong-Ravula adding the conversion.
But Scotland were revived by Townsend's switches in the second half.
Playing with renewed intensity, they battered Fiji into submission as the visitors gradually faded.
Scotland captain Stafford McDowall saluted the fightback, saying: "I'm really proud of the boys. We got loose and let Fiji into the game but we responded and the bench had a really big impact.
"We had to stick to what we have done all year. It was a good sign to see those lads coming on in the second half. I am off to be a groomsman at a wedding now!"
