Turner was all alone in the end zone when he made the grab, which capped a rally during which the Cornhuskers (7-2, 4-1 Big Ten) erased a late 10-point deficit.
Martinez (16-for-36, 159 yards, two touchdowns, three interceptions) pulled Nebraska to within 24-21 with a 35-yard touchdown run.
After the Spartans were forced to punt, Nebraska got the ball back for a last-chance drive, highlighted by Kyler Reed's 38-yard catch-and-run to the Michigan State 20.
On the second try from the MSU 5-yard line, Martinez hit Turner for the game-winner, which put Nebraska into first place in the Big Ten Legends division.
The Spartans (5-5 2-4) were led by Le'Veon Bell, who carried the ball 36 times for 188 yards and two touchdowns.