The penalty try and tries to Visser and Laidlaw allowed Scotland to take a 24-11 lead to halftime and it expanded to 27-11 with a Laidlaw penalty in the 45th minute.
Fiji rushed back into the match with tries to Waisea Nayacalevu (48 minutes) and Metuisela Talebula (65). First five Jonetani Raululu converted Talebua's try on debut to cut the lead to 27-25 as the hosts looked like overwhelming the Scots.
Laidlaw eked out the lead to 30-25 by kicking his fourth penalty, then the fast and powerful Visser touched down for the second time to ensure Scotland escaped with a win.
Visser made his debut only five days after completing a three-year period of residency to qualify for Scotland selection. The Edinburgh winger has been the leading try-scorer in Scottish club rugby for the past three seasons.
"We're pleased to win," Scotland captain Ross Ford said. "We knew how tough Fiji would be. They're very dangerous on counter-attack and they showed us that today.
"They scored some very good tries, they're very dangerous and we had to be on our toes the whole game."
The respective captains, Ford for Scotland and Netani Talei for Fiji, were team-mates until last season at the Edinburgh club.
"I take my hat off to the boys, we played really well," Talei said. "We tried to improve on our last few games and I think we did."
Scotland next plays Samoa in Apia, while Fiji faces Tonga next Saturday in the final match of the Pacific Nations Cup.
In Toronto, Italian first five Kris Burton kicked 20 points, rallying Italy in the second half for a 25-16 victory over Canada.
Burton kicked six penalties and converted Tommaso D'Apice's try in the 50th minute off a driving maul to give the European side the victory in front of a crowd of 12,220.
Both sides used the match to showcase young talent hoping to catch the eye of Italian boss Jacques Brunel and Canadian coach Kieran Crowley.
The Canadians had 11 points from fullback James Pritchard, who kicked three penalties, and converted winger Conor Trainor's first-half try.
The Canadians will entertain Georgia on June 23, the same day the Italians will visit the United States.
-AAP