The outside linebacker stopped Marc Bulger for a 1-yard loss late in the first quarter. The St. Louis Rams quarterback was scrambling after fumbling a snap from shotgun formation and then getting the ball back.
Ware tied the mark set by Denver's Simon Fletcher from Nov. 15, 1992 to Sept. 20, 1993.
Though he shares the NFL record, Ware remains one game shy of the franchise record. The Cowboys recognize Harvey Martin's 11-game streak from 1976-77, a run that ended five years before the sack became an official statistic, as their number to beat.