Nagamura scored in the 75th minute, 10 minutes after entering as a substitute. The goal concluded a 12-pass sequence, including a perfectly placed cross to the penalty spot by Jalil Anibaba.
Sporting KC (3-2-4) got its first win since April 5, while Chicago (3-4-0) had its three-game winning streak halted.
Kansas City goalkeeper Tim Melia, making his first start of the season, had a diving stop of Guly's shot from the top of the 18-yard box in the 13th minute. He pounced on his spilled save in the 18th minute. KC's Dominic Dwyer sailed an open header in the 31st minute and Kevin Ellis headed one off the crossbar 2 minutes later.