Shimonek, ranked second nationally in passing yards per game entering the day, threw touchdown passes to Keke Coutee and Quan Shorts, who led the Red Raiders with 87 and 76 receiving yards, respectively.
The Red Raiders (4-1, 1-1 Big 12) were coming off a narrow defeat to Oklahoma State. The Jayhawks (1-4, 0-2) lost their fourth straight game and have allowed 40-plus points in all four games against FBS opponents.
Texas Tech scored touchdowns on each of its first three possessions, sailing 75, 55 and 58 yards. The last two scores came after fourth-down stops at the Red Raiders 45- and 42-yard lines.
Shimonek completed his first 10 passes and went 16 of 20 in the first half for 161 yards and one touchdown, a 37-yard strike to Shorts, who was standing wide open at the goal line.
The lone first-half touchdown for the Jayhawks came with 19 seconds left in the first quarter on an 18-yard pass from Peyton Bender to Jeremiah Booker. Bender alternated with backup quarterback Carter Stanley in the first half, often on the same possessions, though neither provided Kansas much of a spark.
Kansas used field goals to bridge both halves before Taylor Martin scored on a 2-yard rush with 5:49 left in the third quarter to cut Texas Tech's lead to 35-19.
The Red Raiders responded with 30 straight points, including four touchdowns, the first coming on a 20-yard fumble return by nickel back Justus Parker with 3:44 left in the third quarter. Parker then snagged an interception on the next series.
Tech, which entered the game ranked seventh nationally in turnover margin (plus-7), forced four turnovers.
Desmond Nisby, the Red Raiders' backup running back, scored four times and added 93 yards.
Stanley played the entire second half and finished with 110 yards passing. Bender passed for 146 yards. Each was intercepted once.
Khalil Herbert, who gained 291 yards in the Jayhawks' previous game, carried just five times in the first half but led Kansas rushers with 65 yards on 10 attempts before leaving in the third quarter with an apparent hamstring pull.
The victory was the 11th straight in the series for the Red Raiders, who have not lost to the Jayhawks since 2001. The defeat was the 22nd in the last 23 Big 12 games for the Jayhawks.