Joe Root's England need to win at least two more tests to qualify for the final, which will be held at Lord's in June. New Zealand have already qualified.
Green was caution itself before reaching the half century mark but then accelerated with the declaration in sight, plundering four big sixes before getting a faint edge to a Bumrah delivery.
It ended one of the 32-year-old's most important Test innings, having come to the crease with India in trouble at 64 for three to steer them into a strong position.
It was a determined and crucial knock against a world-class attack by the mild-mannered 32-year-old, who assumed the captaincy from Virat Kohli after he returned home for the birth of his first child.