On the game-winning play, Ajirotutu broke free from defender Sean Smith and made a falling catch as the Chargers (5-6) snapped a three-game losing streak on the contest's eighth lead change.
The decisive score came after Rivers (392 yards, three touchdowns) had engineered a 78-yard drive.
San Diego took the lead just 1 minute after a Dwayne Bowe touchdown reception had handed the Chiefs a 38-34 advantage.
Keenan Allen led the Chargers' receiving corps with nine catches for 124 yards.
Kansas City (9-2) lost a second straight following a 9-0 start despite three touchdown passes by Alex Smith.
Jamaal Charles picked up 115 rushing yards with two scores on 14 carries.