Fiji fought back with scores from Waisea Nayacalevu and Metuisela Talebula but Visser put the result beyond doubt with eight minutes to go.
Scotland got off to a slow start in the searing heat of Fiji and Jonetani Ralulu edged the hosts in to the lead.
But Andy Robinson's men flexed their muscle up-front and a rampaging maul but the Scots in a commanding position. They continued to exert pressure and Laidlaw sniped over from close-range. He then added the extras to hand Scotland the lead.
The Scottish pack continued to cause Fiji problems and they earned a penalty try from South African referee Jaco Peyper to extend their advantage.
Josefa Domolailai hit back for the hosts and the game opened up with both sides demonstrating their attacking intent.
Laidlaw kept the scoreboard ticking over before Nick De Luca sent Visser over with a smart miss pass. The Edinburgh fly-half made no mistake with the conversion as Scotland entered the break 13-points clear.
Fiji fought back following the interval with Nayacalevu and Talebula both crossing the Scottish whitewash.
But Laidlaw steadied the visitors with the boot before Lions contender Visser shrugged off the Fijian defence to grab his second try and put the victory beyond doubt.