After Kim Moon-hwan gave the Black and Gold a 24th-minute lead, Alan Pulido equalized for the Wizards in the 61st minute and Daniel Salloi won it for the hosts three minutes from time.
Story of the match
Sporting Kansas City had a couple of early looks as Salloi's cross was a bit wayward and Jaylin Lindsey fired over with his angled shot and they were made to pay for their early wastefulness.
LAFC opened the scoring in the 24th minute as Jose Cifuentes found Moon-hwan on the wing after the South Korean cut back to his left, beating goalkeeper Tim Melia with a strong effort.
Just before halftime, Black and Gold goalkeeper Tomas Romero missed when trying to punch away a Kansas City corner, but Ilie Sanchez's header behind the Salvadorian was off-target.
Eduard Atuesta had a couple of chances, his first a free kick on the left flank that only just missed and his second that hit the target, but was caught easily by Melia.
A minute after halftime, the Black and Gold nearly doubled their lead as Corey Baird re-directed a Moon-hwan cross that hit Melia in the face. In the 57th minute, Johnny Russell nearly scored before the turning point of the match occurred.
In the 58th minute, LAFC defender Tristan Blackmon hauled down Salloi and was shown a straight red card. The ensuing free-kick saw Gianluca Busio hit the crossbar.
Three minutes later, Kansas City were level. Marco Farfan failed to clear a Salloi cross from the left and Pulido pounced on the rebound, firing a one-timer from the top of the 18-yard box past Romero.
The Wizards were pushing for a winner and nearly found it in the 78th minute when Graham Zusi crossed for Felipe Hernandez and his shot was straight at Romero.
Hernandez then turned provider for the match-winner in the 87th minute as he slipped in Salloi and the Hungarian curled an inch-perfect shot from the far corner to beat Romero.
Carlos Vela had an opportunity in the late stages of stoppage time, but he pushed his shot wide and Sporting Kansas City hung on.
Man of the match: Daniel Salloi
Back from duty with the Hungarian national team at Euro 2020, he kept up his good form that saw him notch two goals and an assist in a midweek victory over Colorado. He drew the red card that led to Pulido's equalizer and scored the match-winner.
Salloi finding his form
After scoring one goal and providing one assist over the previous two seasons, the Hungarian now has seven goals and three assists in 2021. If he keeps up this pace, the Wizards will be a championship contender.
LAFC wasting leads
Bob Bradley has seen his team give away 12 points from winning positions, a major reason why the Black and Gold sit eighth in the Western Conference standings.