Estrada (3-1) struck out eight and scattered six hits and two walks in his sixth quality start of the season for Milwaukee, which has won three straight on the heels of a 1-5 stretch.
Gerrit Cole (3-3) was touched for three runs on seven hits over 5 2/3 innings for the Pirates, who were coming off a successful 6-3 homestand.
Cole and Brewers outfielder Carlos Gomez nearly came to blows the last time these teams met, but there were no noteworthy arguments in the opener of this three-game NL Central series.
Neil Walker's solo homer to deep left-center in the first inning staked the Pirates to an early lead, but it was the only costly mistake made by Estrada.
With the bases loaded and two outs in the home third, Lucroy singled in front of Josh Harrison in right field, and Logan Schafer and Gomez came around as the tying and go-ahead runs.
Harrison did not make a good read on Lucroy's humpback liner, and another minor miscue in the field led a Milwaukee run in the sixth as Jordy Mercer bobbled the transfer of Rickie Weeks' dribbler up the middle, allowing Jean Segura to score.
After Gaby Sanchez's pinch-hit, run-scoring double put runners on second and third with two outs, Smith induced a harmless groundout off the bat of Mercer to keep it a 5-2 game.
Francisco Rodriguez retired the side in order in the ninth to post his 16th save of the season.