Substitute Nacer Chadli slotted home in the fourth minute of injury time to complete a memorable comeback and seal a 3-2 win in the last-16 match in Rostov-on-Don.
Belgium were stunned after Genki Haraguchi and Takashi Inui gave Japan a two-goal advantage early in the second half.
But Jan Vertonghen and Marouane Fellaini pulled them level and Chadli’s winner means they will play Brazil in the last eight.
“That was a test for the team and its character,” said Martinez after Belgium become the first team in 48 years to overturn a 2-0 deficit in a World Cup knockout match.
“We survived it, we have gone through and that is the most important thing.”
“No negatives today, it was about getting through,” Martinez added. “It is a day to be very proud of these players. Keep believing in Belgium.
“In the World Cup you want to be perfect but it’s about getting through, it’s about winning.”
Martinez, the former Everton manager, said: “Let’s congratulate Japan, they played the perfect game. They were clinical on the counter and so solid.
“It was a test of character and you see the reaction of our subs coming on to win the game. It tells you everything about this group of players.”
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