Both LAFC goals came in the second half after longtime Sporting Kansas City goalkeeper Tim Melia departed with a right hamstring injury.
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Maxime Crepeau made three saves and recorded his sixth shutout of the season for MLS-best LAFC (14-4-3, 45 points). Los Angeles is 7-1-1 over its last nine matches.
Sporting Kansas City (5-13-5, 20 points) are languishing in last place in the Western Conference. SKC are 1-4-1 over their last six matches.
Sporting KC have dropped four straight home matches.
Melia made two saves before departing in the 53rd minute. John Pulskamp replaced him and didn't make any saves.
Los Angeles had a 13-11 edge in shots and placed five on target to Sporting KC's three.
Cristian Arango broke a scoreless tie in the 56th minute, shortly after Pulskamp entered the match.
Kwadwo Opoku passed the ball to his right to Arango, who sent the right-footed shot past Pulskamp into the net. It was Arango's team-leading ninth goal of the campaign.
Bale scored his goal in the 83rd minute. Arango fed Bale the ball and he took a step to his left and sent a left-footed shot that caromed off Pulskamp's hands and into the net.
Both teams had a big scoring chance in the first half.
Sporting KC's Erik Thommy, making his MLS debut, sent a right-footed shot just wide right of the net in the 15th minute.