Rallying twice in the second and fourth quarters, Houston held on to put the best-of-seven series at 2-2.
The Warriors got off to a flying start with a 12-0 raid on the visitors who trailed 28-19 by the end of a first quarter that saw Kevin Durant score 10.
But the Rockets fired into action in the second quarter, rallying to go 53-46 up at half-time.
After scoring nine in the first period, Harden brought his first-half total to 24 by the break as Houston rose above the Warriors.
But Curry came back with five triples on the way to 17 as Golden State hit back in the third quarter to retake the high ground.
The Warriors went into the fourth with a sizeable 80-70 lead on the Rockets, but Houston were not down and out.
On a Paul assist Trevor Ariza lifted the Rockets over the Warriors 85-84 with 6:03 left on the clock, Harden kept dunking and the Rockets kept the Warriors at bay.
British boxer Anthony Joshua was courtside and shared a video of him giving Curry an “assist” of his own.
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