Colombia, who are in the last 16 for the first time since 1990 and had never before won three matches at the finals, went ahead with an early Juan Cuadrado penalty at the Pantanal arena.
Martinez struck twice in the second half from passes by halftime substitute Rodriguez and Colombia could even afford to bring on 43-year-old back-up goalkeeper Faryd Mondragon who became the oldest player to take the field at a World Cup.
Rodriguez, one of eight players rested from the starting lineup by coach Jose Pekerman after the 2-1 win over Ivory Coast that secured Colombia’s second-round berth, wrapped up the win with a delightful fourth goal.
“I think what James has done today simply demonstrates what we expected from him. From the beginning he has shown he is very well physically and very fit and we can expect a lot from him,” Pekerman told reporters.
Japan equalised just before halftime with Shinji Okazaki’s fine diving header but fell apart near the end of the match.
“It was a difficult first half, we had lots of opportunities to score but found that their strong defence did a lot of damage,” Martinez said.
“In the second half, we were a lot better and were able to show the kind of play we had in previous games,” added Martinez, playing in place of Teofilo Gutierrez.
Greece beat the Ivory Coast 2-1 in the other game in the section to set up a last 16 clash with Costa Rica while Japan finished bottom of the group with one point.
The Japanese suffered their worst defeat at a World Cup since they lost by the same score against Brazil in 2006 in Germany but still contributed to an entertaining and exciting last match at the Cuiaba venue.
Colombia took the lead after 17 minutes when Adrian Ramos was brought down by centre back Yasuyuki Konno as he raced into the box on the left and Cuadrado calmly converted from the spot at the Pantanal arena.
Japan got a brilliant and well deserved equaliser on the stroke of halftime after enjoying more of the possession in the first half and constantly probing Colombia’s reserve central defensive pairing of Carlos Valdes and Eder Alvarez Balanta.
Keisuke Honda sent a low, left footed cross in from the right corner of the box and Okazaki got in front of Valdes to steer a diving header past goalkeeper David Ospina just inside the near post.
With Colombia’s main marksman Radamel Falcao missing the tournament through injury and Teofilo Gutierrez rested, Martinez underlined Colombia’s depth of striking talent when he put them ahead again 10 minutes into the second half.
Right back Santiago Arias cut inside and passed to Rodriguez who laid the ball off to his left from where Martinez shot home across the diving Eiji Kawashima.
With Rodriguez dictating the pace of Colombia’s counter-attacks, Martinez scored his second eight minutes from time as Japan threw bodies forward looking for an equaliser.
Rodriguez, fed by Martinez, then scored with a darting run and chip over Kawashima to complete the rout with his third goal of the tournament in the dying minutes.
Asked about facing Uruguay, Martinez said: “It’s going to be a really tough game, a fight, and we just have to maintain our focus and continue to work like we have.”
Japan should have scored a second equaliser in the 66th minute but striker Yoshito Okubo blasted right back Atsuto Uchida’s cross over the bar from right in front of goal with Ospina at his mercy.
The Japanese enjoyed more possession but were overcome by the power of the counter-attacks unleashed by Colombia who missed several other chances.
“I have always thought and said that Japan were a really good team and we saw that when they played the Asian qualifiers,” Pekerman said.
“Often today they played a very good game, they didn’t get the results they expected. They were facing a good Colombian team.”
Pekerman sent Mondragon on for the final five minutes, allowing the veteran of the 1994 and 1998 finals to make a last mark on the event as the oldest man to play in a World Cup.
Mondragon surpassed Cameroon’s Roger Milla, who played at the age of 42 in the 1994 World Cup in the United States, and made a fine block in stoppage time to keep out a point-blank effort by Japan substitute Yoichiro Kakitani.
The match ended with fans that packed the stadium chanting the names of their heroes Martinez, Rodriguez and Mondragon.
(Additional reporting by Mary Milliken; Editing by Ken Ferris)