Bouanga found the net unassisted in the 40th minute to send LAFC (4-0-2) into halftime with a 1-0 lead.
Bouanga added another unassisted goal in the 58th minute to push the advantage to 2-0 and put the finishing touches on the victory with a score in the 68th minute. Sergi Palencia and Carlos Vela picked up assists.
LAFC had a commanding 26-9 advantage in shots but just a 6-4 edge in shots on goal.
Eldin Jakupovic turned away four shots to earn a clean sheet for LAFC. Brad Stuver had three saves for Austin (2-3-1).
Austin beat LAFC twice during the regular season last year but lost 3-0 to the eventual MLS Cup champion in the Western Conference final. Austin falls to 1-6-0 in its last seven road matches dating to last season.
LAFC is 13-1-1 at home in its last 15 matches, including the playoffs.
Austin returns home to host the Vancouver Whitecaps on Saturday. LAFC play on the road against the Galaxy on April 16.