It was a meeting of unbeaten teams and the Blue Devils (7-0) prevailed with Mason Plumlee scoring 21 points and gathering in 16 rebounds.
Duke fell behind by 10 points early in the second half, but took the lead for good at 56-53 on a 3-pointer by Ryan Kelly with 5:42 to play. The Blue Devils had a 66-64 advantage when Cook began his parade to the foul line.
Rasheed Sulaimon added 17 points for the Blue Devils, Kelly had 15 and Cook scored 12.
Ohio State stayed in the game by grabbing 16 offensive rebounds to make up for a 33-percent shooting effort.
DeShaun Thomas led the Buckeyes (4-1) with 16 points.