The loss denied Japan a chance to move straight to the knockout stage.
Fuller's score came on Costa Rica's first shot on target in the match.
Costa Rica, which lost to Spain 7-0 in its Group E opener, now has an improbable chance to advance. With two matches down, Costa Rica and Japan have three points each. Spain also has three points and will meet Germany, which lost its opener to Japan, later Sunday.
Japan controlled the ball 57 percent of the match and had three shots on goal, all stopped by Costa Rica goalkeeper Keylor Navas. The Japanese also couldn't take advantage of five corner kick opportunities.
Morocco 2, Belgium 0
Abdelhamid Sabiri scored in the 73rd minute and fellow substitute Zakaria Aboukhlal also tallied in second-half stoppage time to lift Morocco to victory in its Group F match.
Sabiri opened the scoring after curling in a free kick from near the corner flag that surprised Belgium goalkeeper Thibaut Courtois (two saves).
Aboukhlal provided insurance with a goal on the counterattack and Munir El-Kajoui made three saves for Morocco, which has four points -- one more than Belgium.
The Belgians were unable to take advantage of any of their nine corner kicks.
Croatia 4, Canada 1
A quick-strike goal against Croatia did not hold up and Canada crashed out of theWorldCupwith a second straight setback in Group F.
Alphonso Davies' leaping header gave Canada the lead in the second minute and marked the first goal scored by the men's national team inWorldCuphistory.
Croatia grabbed the lead with first-half goals by Andrej Kramaric in the 36th minute and Marko Livaja in the 44th.
Kramaric scored again in the 70th minute and Lovro Majer capped the Croatians' win with a goal during second-half stoppage time. Croatia and Morocco lead Group F with four points.
Spain 1, Germany 1
An 83rd-minute goal by substitute Niclas Fullkrug salvaged a crucial point for Germany in a Group E draw against Spain.
Alvaro Morata broke the deadlock for Spain in the 62nd minute.
Unai Simon made three saves for Spain, including an impressive stop against Joshua Kimmich in the 56th minute.
Spain leads the group with four points, followed by Japan and Costa Rica with three each and Germany with one.