The Central Americans finished second in Group D and will face gold medal favourite Brazil in the last eight on Saturday.
"I am very, very proud of the players," Honduras coach Luis Suarez said. "Over the last seven months they worked very hard. The most important thing is the attitude, union and friendship."
Japan, which had already qualified for the last eight, secured top spot in Group D with the draw at the City of Coventry Stadium and set up a quarterfinal clash with Group C runner up Egypt.
Honduras had the better of the opening exchanges, though, winning three corners in the first 12 minutes.
The Japanese had the first scoring opportunity after a quarter of an hour when Yuki Otsu broke forward and shot from far out, forcing a comfortable save from Jose Mendoza in the Honduras goal.
Bengston's near goal came seven minutes before half time, when he fired in a shot on the turn inside the box. Gonda was up to the task and dived low to save.
Two minutes later Gonda was called into action once again, this time stopping a long-range free kick from Maynor Figueroa.
Japan continued to have the majority of possession in the second half, but had little to show for it, with Takashi Usami's 80th minute free kick the only effort to bring a real save from Mendoza.
"Our main target was to come in first place (in the group) and we achieved that," Japan coach Takashi Sekizuka said. "We are playing as one and showing what we can do."