Harden led the way with his play-off best total, while Dwight Howard rebounded 26 times as the Mavs were beaten 130-128 in Houston.
The Washington Wizards are also on for a first-round sweep after they beat the Toronto Raptors 106-98 in the capital.
Veteran Paul Pierce landed a vital three-pointer to seal the deal late in the game, with John Wall having returned 19 points and 15 assists.
Friday’s other game saw defending champions San Antonio go 2-1 up on the Los Angeles Clippers with a 100-73 rout.
The Spurs lost the first game on the series but are ahead now, defensive player of the year Kawhi Leonard top-scoring with 32.