Visser's second try in the 72nd minute took the sting out of a Fiji fight-back after the hosts had rallied from 24-11 down at half-time to just a two-point deficit with 12 minutes remaining.
Laidlaw's third penalty had given Scotland some breathing space at 30-25 before Visser crossed for his second try, which Laidlaw converted to cap the scoring.
Andy Robinson's side beat Australia 9-8 in a rain-drenched game in Newcastle on June 5, a result that caused consternation amongst Wallabies supporters as they prepared for their series against Six Nations champions Wales.
Laidlaw, who booted three penalties in that victory over Australia, finished the match against Fiji with 22 points from four conversions, three penalties and his side's first try after some good build-up work by the outside backs.
Fiji fly-half Jonetai Ralulu had given the home side an early 6-0 lead before Laidlaw's try and Scotland were never behind for the rest of the match.
Scotland were awarded a penalty try to give them a 14-6 lead before replacement forward Josefa Domolailai grabbed a try to drag the hosts to 14-11.
Edinburgh winger Visser, who was born in the Netherlands and qualifies for Scotland through residency rules, then scored his first try, which Laidlaw converted before he added a penalty to take the visitors into the break with a comfortable 24-11 lead.
However, Fiji were fired up after the break and while Laidlaw slotted his second penalty the hosts got back into the game with some good work by their forwards resulting in winger Waisea Nayacalevu crossing.
Fiji threatened to take control when replacement Metuisela Talebula finished a sweeping movement to get them to 27-25 after Ralulu converted.
Laidlaw, however, extended the lead with his third penalty before Visser crossed for his second try two minutes later to seal the Scottish win.