The Redhawks' Kyle Snyder threw three touchdown passes, all in the fourth quarter, the last of them a 26-yard pass to Paul McRoberts on fourth down with 1:33 left. The Redhawks (1-1) tried an on-side kick, but Harwell hopped on the loose ball, allowing Kansas to run out the clock.
Snyder wound up throwing for 269 yards, though he also tossed three interceptions.
Dexter McDonald had two of them for the Jayhawks, who are in their third year of Charlie Weis's massive rebuilding project. And if there isn't significant progress this season -- they've won four games his first two -- he may not be around for the rest of his five-year contract.
The Redhawks, who beat football newcomer Missouri Baptist 77-0 a week ago in coach Tom Matukewicz's debut, found it hard to overcome themselves. They were penalized 14 times for 174 yards, the abundance of yellow flags often scuttling some promising drives.
In typical Kansas fashion, a stretch of very good play was followed by some very bad.
First was the good, a 24-point opening quarter that was the Jayhawks' best 15-minute period since they scored 34 points in the fourth quarter of a game against Colorado on Nov. 6, 2010.
After taking a 3-0 lead, Matthew Boateng's pick gave the Jayhawks good field position. Three players later, Avery took an option pitch over the left side from 10 yards out for a score.
Cozart added a 6-yard TD pass to Harwell later in the first quarter, and his 10-yard pass to the former Miami of Ohio star moments later gave the Jayhawks a 24-0 lead.
Then came the bad, a second quarter spent mostly going backward.
The Redhawks sacked Cozart to foil one drive, forced a three-and-out on the next, and then caused a comical fumble to end the half. While players from both teams jumped into a pile near midfield, the loose ball squirted away without anybody noticing. Kansas right tackle Damon Martin finally did, but only after staring dumbfounded at the ball for several seconds.
The Jayhawks managed just 13 yards in the second quarter.
The Redhawks finally got going in the third, blocking Matt Wyman's 49-yard field goal attempt and returning the ball to the Kansas 22. Jackson scored four plays later to make it 24-7.
Sensing some momentum, Southeast Missouri State appeared to recover an onside kick, only to be flagged for being offside. Kansas wound up with good field position on the ensuing kick, and Cozart hit Tony Pierson for a 67-yard TD pass that briefly restored some order.
Southeast Missouri State refused to go quietly.
Snyder hit Spencer Davis with a 37-yard TD strike early in the fourth quarter, and then found McRoberts from 68 yards out to make it 34-21 with 10:39 left. But by the time he hit McRoberts to make it a one-possession game, the Redhawks had nearly run out of time.