Scotland led 24-11 at half-time in blistering heat in Lautoka thanks to tries from Greig Laidlaw and Visser which sandwiched a penalty try.
Fiji, for whom Josefa Domolailai crossed in the first period, fought back after the break and got scores from Waisea Nayacalevu and Metuisela Talebula, which put them within two points at 27-25 with 12 minutes to play.
But Visser claimed his second try eight minutes from the end and Laidlaw converted, taking his personal tally to 22, to settle Scotland nerves.
Fiji dished out some huge hits early on and with Scotland failing to control either the ball or their discipline, they soon conceded early penalties, two of which Jonetani Ralulu banged between the posts though he also missed one from inside his own half.
Scotland had hardly seen the ball inside the first quarter but when they did get it, they at last managed to show what they have been threatening all season with a flowing move that sent the ball both ways before Visser, on his debut, was sent clear on the left.
He had support inside and though Al Kellock was caught a yard short, Laidlaw was there to pick up and dive over.
Two minutes later, they were at it again. This time is was Max Evans, the wing, who was stopped short. Fiji killed the ball, and from the resulting penalty Scotland went for the scrum, drove almost to the line and were awarded a penalty try when Fiji illegally kicked the ball out inches before it crossed the line.
Laidlaw converted both but it was not the start of a Scottish try-fest, the next score coming at the opposite end of the park where scrum-half Nikola Matawalu nipped through the side of a ruck and slipped the ball inside to Domolailai, the replacement flanker, who was waiting to crash over.
Ralulu missed the conversion, and Scotland were able to stretch their lead with a Laidlaw penalty before the moment all Scottish fans had been waiting for - Visser's first try in Scotland colours.
It was set up by another immense scrum drive, before the ball was released wide and Nick De Luca gave him the sort of space that he was never going to squander.
Comfortably ahead at the break, the question was whether Scotland would wilt in the heat or press home their advantage.
A brilliant solo break from Matawalu threatened a Fiji fightback, but Evans got back to save his side before Laidlaw extended the lead with a penalty.
They needed it too as Watisoni Votu made a superb solo break to create the threat, finished on the other side of the field by Nayacalevu.
Suddenly Fiji had their tails up and the crowd started to get into the game, rousing the players who started to show all the fluency and imagination for which they are famed.
Somehow Scotland held firm for a while but the brilliance of the Fijians could not be denied.
A high kick was tapped back to support and centre Vereniki Goneva raced through the Scotland defence before finding Talebula, the replacement full-back, for a run to the line from a move that had started on their own 22.
For all their attacking flair, Fiji can still self destruct, and their failure to handle a high kick under pressure from Matt Scott presented Visser with simple run-in for his second try, to wrap up a nervy but satisfying win for Scotland.
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