Dickerson stroked a two-run single to produce a 2-2 tie in the first inning and pushed across another run in the next frame on a fielder's choice and the Brewers went on to sweep a three-game set with the Pirates.
Ryan Braun homered and went 4-for-4 and George Kottaras added three hits and an RBI for the victors.
Dave Bush (7-11) held Pittsburgh to four runs on seven hits in seven innings of work to collect the win.
Pirates starter Charlie Mortion (1-10) was pounded for eight runs on nine hits in a 3 1/3-inning start.
Delwyn Young, Neil Walker and Ryan Doumit all homered for Pittsburgh.