Harden top-scored for Houston with 35 points.
Kevin Durant led Golden State with 35 points while Curry added 18 points.
«We just made them shoot a few tough shots,» Durant said. «I thought our defence won us the game.»
Bogut started on the bench and only spent four minutes on the court in the second quarter, scoring two points and grabbing three rebounds.
Warriors coach Steve Kerr went with veteran guard/forward Andre Iguodala in the starting line-up alongside regular starters Curry, Durant, Klay Thompson, and Draymond Green.
Kerr also went with back-up centre Kevon Looney as his first big man option off the bench to combat the smaller, mobile Rockets who started 208cm tall Clint Capela at centre.
The Warriors are the west’s top seed and the Rockets the fourth.
Game two is in Oakland on Tuesday.
The Celtics crushed the Bucks 112-90 in game one of the Eastern Conference semi-final series in Milwaukee.
Baynes played a significant role in taming the Bucks’ NBA MVP candidate Giannis Antetokounmpo early, with the Greek star held to just eight points in the first half while the Queenslander and Celtics forward Al Horford took turns guarding him.
Antetokounmpo ended with 22 points and eight rebounds with the fourth-seeded Celtics effectively restricting the usually rampaging Milwaukee’s fast breaks and points in the paint.
Baynes rolled his left ankle late in the third quarter while scrapping for a rebound.
The Australian centre, who came off the bench for four points, has had a tough season missing 31 games with a broken hand as well as hamstring, foot and ankle injuries.
Boston coach Brad Stevens said Baynes was cleared by the team’s medical staff to return to the court but he kept him on the bench.
«He’s rolled it a few times in the last month, so we’ll see how he feels,» Stevens said.
«He’s obviously an important part of our team.»
Game two is in Milwaukee on Tuesday.
Boston’s Melbourne-born All-Star Kyrie Irving led the way with 26 points, seven rebounds and 11 assists while Horford had 20 points and 11 rebounds.