Uggla unleashed a three-run blast in the third inning for a 4-1 Atlanta lead and added a solo shot in the fifth to cap the scoring.
In between, Freddie Freeman added a bases-clearing double for the National League East-leading Braves, who split a four-game series at Dodger Stadium.
Minor (8-2) ran his unbeaten streak to eight starts, taming Los Angeles on one run through six innings. The left-hander issued three walks and struck out six.
Matt Magill (0-2) was lit up for seven runs (three earned) on four hits over 3 2/3 innings after handing out six walks.
Skip Schumaker drove in the Dodgers' only run.