Payne sank 10 of 15 shots from the field, including four of five from 3-point range, and went 17-for-17 at the free-throw line. He received a standing ovation from the Spokane Arena crowd of 10,862 when coach Tom Izzo removed him from the game in the final minute.
The Spartans (27-8) face 12th-seeded Harvard (27-4) on Saturday, with the winner advancing to the East Regional semifinals in New York. The Crimson defeated fifth-seeded Cincinnati 61-57 earlier Thursday.
Michigan State is playing in the NCAA Tournament for the 17th consecutive year and the 28th time overall. Delaware, making its first NCAA appearance since 1999, fell to 0-5 all-time in the tourney.
Senior guard Devon Saddler, Delaware's all-time leading scorer, paced the Blue Hens (25-10) with 21 points. Backcourt partners Davon Usher (20) and Jarvis Threatt (14) also scored in double figures for Delaware.
The Spartans shot 52.6 percent from the field, and the same percentage on 3-pointers (10-for-19). Delaware shot 37 percent from the floor. The Blue Hens nailed 40 percent of their 3-pointers (8-for-20).
Payne scored 23 points in the first half to help the Spartans build a 44-33 lead at the break. Payne sank six of eight shots from the field, including all four of his 3-point attempts, and went 7-for-7 at the free-throw line.
Payne's 3-pointer gave the Spartans a 36-18 lead with 6 1/2 minutes left in the first half, but the Blue Hens scored the next 11 points. Saddler and Harris both scored 12 points each in the first half.
Michigan State outrebounded Delaware 22-10 and forced 10 turnovers in the opening half.