The Suns won for the fourth time in five games and dropped the Clippers back into a tie with the Los Angeles Lakers for first place in the Pacific Division.
Phoenix overcame an 11-point deficit in the first half and won despite shooting only 34 percent from the field.
Chris Paul made a 3-pointer for the Clippers with 29 seconds left to bring them within two, but Dudley's two free throws put the Suns in control.
Paul then missed a 3-pointer with 10 seconds remaining to wipe out the Clippers' last hope and Grant Hill scored on a layup 3 seconds later to put Phoenix in front by six. A 3-point play by the Clippers' Blake Griffin in the final second accounted for the final score.
Dudley led the Suns with 22 points and Marcin Gortat had 18 points and 14 rebounds.
Griffin had 17 points, but made only 6-of-19 from the field.