Curry hit three 3-pointers and was perfect in six trips to the free-throw line to lead the Blue Devils, who got hot in the second half and outscored Kentucky (1-1) 42-37.
Mason Plumlee added 18 points for Duke and Ryan Kelly and Rasheed Sulaimon added 10 apiece.
Kentucky was led by Alex Poythress with 20 points and eight rebounds. Archie Goodwin and Nerlens Noel both had 16.
The Blue Devils, who led 33-31 at halftime, shot 45.6 percent from the field to Kentucky's 49 percent. Duke sank eight 3-point shots to Kentucky's four.