Watched by Japan's Emperor Akihito and Empress Michiko, and a record rugby crowd of 31,000 in Tokyo, Laidlaw kicked four second-half penalties to give the tourists a victory they barely deserved.
Beaten 26-13 last weekend, Japan led 13-9 at halftime, punctuated by a breathtaking try from scrum-half Kaito Shigeno, who began and finished a superb, coast-to-coast move after 21 minutes.
The Scots, who inflicted Japan's only loss of their astonishing World Cup under Eddie Jones last year, struggled to get a foothold against some tenacious tackling from the Brave Blossoms.
Yu Tamura kept the scoreboard ticking over for Japan with three penalties, the third giving Japan a 16-9 advantage early in the second half.
But Scotland coach Vern Cotter, who had rested Laidlaw to give Henry Pyrgos his fifth international start at scrum-half, turned to his trusty skipper to rescue the situation after 50 minutes.
Laidlaw's right boot quickly brought Scotland to within a point and they took the lead on 71 minutes with another huge kick from the Gloucester star, who popped over his fourth penalty two minutes from time.
Japan interim coach Mark Hammett made just two changes from last week's side but was missing key players with injury, with skipper Michael Leitch and Ayumu Goromaru watching on in suit and tie.
The Asian champions famously stunned South Africa in their opening game at the 2015 World Cup but Scotland took the wind out of their sails with a crushing 45-10 victory in their next match.
Japan, who subsequently beat Samoa and the United States but narrowly missed out on a place in the knockout phase, host the next World Cup in 2019.