The Wildcats (30-5) scored 41 points in the first half and weathered a second-half Delaware (32-4) rally to advance to the Elite Eight and a meeting with the winner of the Maryland-Connecticut tilt.
A'dia Mathies chipped in 16 points and 10 rebounds for Kentucky.
Elena Delle Donne led Delaware with 33 points and nine boards in her college finale. She finished her career as the fifth-leading scorer in NCAA women's basketball, passing standouts Cheryl Miller, Maya Moore and Chamique Holdsclaw.
Kentucky led 41-27 at halftime but the Blue Hens dominated the paint in the second half to cut the gap to 62-60 with 2:47 to play. Kastine Evans then nailed a 3-point shot that secured the win for Kentucky.