Phoenix moved into a tie with Utah for the eighth and final playoff berth in the Western Conference. Both teams have three games remaining and they will meet next Tuesday in Salt Lake City.
After Nash's two free throws gave Phoenix a 91-90 lead, Sebastian Telfair blocked a layup attempt by the Clippers' Chris Paul. Los Angeles came up with the offensive rebound, but Mo Williams missed a 3-point attempt with 2 seconds to go.
Channing Frye was fouled immediately after grabbing the rebound and his two free throws accounted for the final score.
The Clippers missed a chance to move into a first-place tie with the Los Angeles Lakers in the Pacific Division.
Jared Dudley led Phoenix with 18 points. Marcin Gortat had 14 points and 14 rebounds for the Suns.
Paul had 19 points for Los Angeles, but he made only 6-of-19 from the field.