Rookie Jayson Tatum scored 24 points to lead five Boston players in double figures as the Celtics took a 3-2 lead in the best-of-seven Eastern Conference finals.
P.J. Tucker had 16 rebounds and Clint Capella grabbed 13 for the Rockets, who host game five of the best-of-seven Western Conference final on Thursday in Houston.
The 33-year-old superstar made 17-of-28 shots from the floor, 1-of-4 from 3-point range, and 9-of-13 free throws while contributing five rebounds, three assists and two steals to spark a crucial home triumph.
The reigning champions surpassed the mark of 15 straight home playoff wins set by the Chicago Bulls from April 27, 1990 to May 21, 1991.
Curry, targeted relentlessly by Houston as the Rockets levelled the series in game two, overcame a sluggish first-half display to lead a Warriors blowout in game three.
The Cavaliers, trying to return to the NBA Finals for a fourth straight year, insisted after two double-digit defeats in Boston that things would be different on their home floor.
James, who at 33 produced one of the best seasons of his outstanding career, is in the running for a fifth career MVP title.
Houston's Most Valuable Player candidate James Harden scored 27 points and pulled down 10 rebounds.
Peru hosted the 2018 start in January when the race started in the capital Lima and crossed Bolivia before finishing in Argentina's Cordoba.
The Boston side have never blown a 2-0 series lead in the postseason.
Kevin Durant scored 37 points and Klay Thompson added 28 for the Warriors to capture the best-of-seven Western Conference final opener.
The Rockets, sparked by NBA scoring champion James Harden, were 2-1 against Golden State in the regular season and went a league-best 65-17 overall.
The Cavaliers visit Boston on Sunday for the opening game in the best-of-seven Eastern Conference Finals, with the winner to face either regular-season pace-setter Houston or defending champion Golden State in the NBA Finals.
Despite the disappointment of their playoff exit, the young 76ers were sounding a note of optimism after an overachieving season.
The Celtics advance to the NBA's final four to face LeBron James and the Cleveland Cavaliers for the second straight year for a berth in the finals.