Rodney McGruder added 15 for the Wildcats (11-2). Kansas State dominated the boards and shot 50 percent to South Dakota's 32.7 percent.
Karim Rowson led the Coyotes (6-9) with 14 points. Juevol Miles, who transferred from Kansas State to South Dakota, had 13 points.
Trailing by 12 points, South Dakota started the second half with an 11-5 burst. Trevor Gruis' two foul shots with 15:13 to play cut the deficit to six points.
Kansas State led 44-39 with under 8 minutes left before taking control with a 12-0 run, starting with a three-point play by Jordan Henriquez.
Williams was then intentionally fouled on a breakaway layup and made both free throws. Shane Southwell followed with a 3-pointer and two more baskets.
Kansas State jumped to an 11-2 lead, with its size in the post continually posing a problem for South Dakota. The combination of 7-footers Henriquez and Adrian Diaz made it difficult for the Coyotes to score inside early.
The Wildcats led 30-18 at halftime.