Harden scored 25 points in the second half.
Michael Beasley's 3-pointer with 2 seconds left in the third gave the Suns the lead going into the fourth quarter, 77-76.
The Suns (22-41) used a 16-0 second-quarter run that spanned 5 minutes, 40 seconds to take a 50-46 halftime lead. During the run, Dudley gave the Suns their first lead, 37-35, with a short jumper.
Back-to-back 3s by Dudley made it 45-35, and it wasn't until Parsons made a free throw for Houston with 3:17 left in the quarter that the run ended.
The Rockets led by as many as eight points in the first quarter. The teams face each other three more times before the regular season ends in mid-April.
NOTES: Suns center Jermaine O'Neal remains away from the team after leaving to be with his daughter, who had heart surgery earlier in the week. He missed his third straight game. Suns coach Lindsey Hunter said he's exchanged phone messages with O'Neal and that all indications are that the surgery went well. "My prayers are with his family," Hunter said. "I think family is the most important thing. ... The Suns wore their "Los Suns" orange jerseys and warmup tops in recognition of the NBA's "Noche Latina," the league's 16-game tribute to its Latino fan base. Seven teams in large Hispanic markets are taking part. Harden got a loud applause from the crowd, as he played his college ball at nearby Arizona State University. ... Rockets guard Jeremy Lin left the game briefly with a cut lip after a third-quarter collision on which he was called for an offensive foul. He was examined by a trainer and eventually returned.