Goalkeeper Donovan Ricketts earned his third shutout in the Galaxy's first league win at home over the Dynamo (8-3-4).
Dunivant scored his first goal in three years by using his left foot to deflect Mike Magee's 14-yard shot inside the right post from seven yards out.
Magee began the scoring sequence by playing a short corner kick to Eddie Lewis, receiving Lewis's return pass and faking the Dynamo's Brian Mullan out of position.
Ricketts finished with three saves in his first shutout since May 2. He made his biggest in the 90th minute, when he blocked Cam Weaver's low 5-yard shot off Stuart Holden's corner.
Bryan Jordan had a chance to score for the Galaxy (3-3-9) on a breakaway in second-half stoppage time, but goalkeeper Pat Onstad blocked his point-blank shot near the penalty spot.
Galaxy defender Omar Gonzalez left in the 62nd minute with cramps in his hamstrings.
Los Angeles won without forward Landon Donovan, who scored a goal for the United States in its 3-2 loss to Brazil in the FIFA Conferations Cup final in South Africa.
Dynamo midfielder Ricardo Clark also played in that game.
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