The distraught Japanese, who went 2-0 up with goals by Genki Haraguchi and Takashi Inui early in the second half, fell to the turf after Chadli poked the ball home from Thomas Meunier’s cross four minutes into added time at the Rostov Arena.
Belgium, who had taken a conservative approach to the game, got back into contention with late goals by Jan Vertonghen and substitute Marouane Fellaini before Chadli struck.
“That’s what happens in the World Cup. It was a test of character and a test of the team. It tells you everything about this group of players,” said Belgium coach Roberto Martinez.
One of the most exciting teams of the group phase, Belgium faced an embarrassing exit but finally clicked to become the first team to win a World Cup knockout game from two goals down since West Germany beat England 3-2 after extra time in 1970.
With the scores locked at 2-2 and the game in stoppage time, Keisuke Honda’s 30-metre free-kick was parried away by Courtois.
The keeper then launched the ball to Kevin De Bruyne, who made a beeline through the field before finding Meunier, whose cross was met by Chadli for the winner.