Randy Foye had 28 points with a career-high eight 3-pointers while Blake Griffin had 15 points and 16 rebounds for the Clippers, who last won six in a row in March 1992.
Foye was 10 of 19 from the floor, including 8 of 15 from 3-point range.
Dirk Nowitzki led Dallas with 19 points.
