The two-time defending champion Warriors trailed by 18 in the second quarter, but with Green driving them they produced another dominant third period to seize control of the contest and the series.
It was another huge performance from Curry, who has stepped into the void created by Kevin Durant's injury absence.
Giannis Antetokounmpo made 24 points, and also hauled down 14 rebounds with six assists.
Klay Thompson added 26 points and provided an imposing defensive presence as the Warriors shrugged off the absence of injured superstar Kevin Durant.
It will be the first time a pair of brothers have played each other in a conference final.
CJ McCollum carried the load as Portland Trail Blazers also won a tight game seven over Denver Nuggets to progress.
The two-time defending champions punched their ticket to the NBA's final four for the fifth consecutive season and eliminated the Rockets from the postseason for the fourth time in five years.
Jimmy Butler scored 25 points while Ben Simmons and Joel Embiid also had spirited performances as the 76ers survived to fight another day with a 112-101 win over the Toronto Raptors in game six.
The Warriors said he had suffered a right calf strain but Durant is expected to undergo an MRI scan to Thursday to determine the full extent of the damage.
It was a quick turnaround from the Trail Blazers' 140-137 quadruple overtime victory in Friday's game three.
It was just the second quadruple-overtime playoff game in NBA history, joining a 1953 game between the Boston Celtics and the Syracuse Nationals.
Joel Embiid finished with 33 points 10 rebounds and five blocked shots for his 12th double-double in 15 career playoff games.
The event will take place May 30-June 2 at Stade Louis II in Monaco.
McCollum led the scorers with 20 points in a hard-fought 97-90 victory on the road at Denver's Pepsi Center to tie the best-of-seven series at 1-1.
Giannis Antetokounmpo and Milwaukee Bucks bounced back to knot their playoff series with Celtics at one game apiece.