The Reds also had only four hits, one a home run by Kyle Farmer.
All three Wong home runs landed in the right-field bleachers and the first two came after Reds pitchers issued walks.
The first one came in the second inning after Greene walked Hunter Renfroe on four pitches.
The second one came in the sixth inning after relief pitcher Dauri Moreta walked Rowdy Tellez on a full count.
The third one was a solo rip, a two-out shot off relief pitcher Joel Kuhnel. The home runs were numbers 13, 14 and 15 for Wong.
The Reds threatened Woodruff in the second inning with back-to-back one-out singles by Donovan Solano and Alejo Lopez, but Matt Reynolds and Michael Siani struck out.
Siani? He made his major-league debut and was the 66th different player to appear in a game for the Reds this season.
T.J. Friedl walked on four pitches to open the sixth but made another of the mounting baserunning blunders concocted by the Reds this season. He strayed too far off first base on Jonathan India’s fly ball to right and was doubled off first.
Farmer, 1-for-23, followed with his home run that merely cut Milwaukee’s lead from 4-0 to 4-1.
After Woodruff departed, the Brewers bullpen shut it down. Pete Strzelecki, Matt Bush and Jake Cousins held the Reds to no runs, one hit and one walk over the final three innings.
Every victory is important for the Brewers as they pursue a National League wild card spot and are 1 1/2 games behind the San Diego Padres for the final spot.
Meanwhile, the Reds have lost 11 of their last 14 games and own 91 defeats.