Mario Chalmers had 18 points and Darnell Jackson had 13 points and nine rebounds for the Jayhawks (18-0, 3-0 Big 12), off to their best start since a school-record 22-0 in 1996-97. Rush, the younger brother of former Missouri star Kareem Rush, went 9-for-14 from the line and finished with 13 points and nine rebounds.
Kansas has won four straight and eight of 10 in the hotly contested series, winning this one despite 40 percent shooting - only their fourth game below 50 percent. The Jayhawks compensated by making a season-high 28 free throws in 45 attempts, and outrebounding Missouri 52-43.
The latest matchup attracted a crowd of 15,061, the first sellout crowd of the season at Mizzou Arena, which had been averaging just over 6,000.
Stefhon Hannah had 23 points for Missouri (11-7, 1-2), which has lost two straight since beating then-No. 12 Texas a week ago. The Tigers ended a 13-game home winning streak, including 10 this season, although they made the Jayhawks work for it.
DeMarre Carroll had a season-low five points on 2-for-11 shooting, but had 12 rebounds.
Kansas, which entered the game averaging a nation-best 25-point winning margin, never led by more than eight and needed an 8-for-12 finish at the free throw line.
A pair of baskets by Jason Horton in an 18-second span cut Missouri's deficit to 66-63 with four minutes to go, but the Tigers came up empty on their next four possessions and Rush hit a pair of free throws for a 70-63 lead after an intentional foul by Matt Lawrence, who collared him as he swept in for a fast-break dunk with 1:04 left.
Missouri jumped to a 7-1 lead in the first two minutes, taking advantage of an 0-for-4 start by Kansas, and kept the momentum most of the half. Kansas' two-point lead at the break at 38-36 matched its biggest cushion.
Rush was scoreless the first 10 minutes and was 1-for-7 in the half.
Missouri junior Vaidotas Volkus got a surprise start, the first of his career, at forward in place of struggling Leo Lyons and had eight points on 4-for-5 shooting. Lyons had 10 points and five rebounds before fouling out with 4:52 to go.