After three scoreless innings, Greene gave up three runs to open the fourth — single by Rowdy Tellez, single by Andrew McCutchen and a three-run first-pitch home run to Hunter Renfroe.
And the Brewers made it 4-0 in the fourth when Willy Adames reached the right field seats with a home run.
The Reds used the same modus operandi to get back into the game against Milwaukee left-hander and Kent State product Eric Lauer.
Mike Moustakas pulled a one-out single to right in the fifth and Albert Almora Jr. reached the left field seats for a home run that drew the Reds to within 4-2.
It took the first two pitches of the sixth inning for the Reds to tie it, 4-4. Brandon Drury and Tommy Pham hit back-to-back first-pitch home runs.
Lauer is susceptible to the long ball. He has given up 16 homers this season and been touched up for three home runs in a game three times.
Greene, though, needed 97 pitches to cover five innings and manager David Bell went to the bullpen.
Jeff Hoffman walked his first batter in the sixth, then retired the next three. With a string of left-handers due up, Bell went to lefty Ross Detwiler with one out in the seventh.
He retired lefty Jace Peterson, but lefty Keston Hiura left the playing field with a home run for a 5-4 Brewers’ lead that stood the test of time.
The Reds did put two runners on base with one out in the ninth but stranded them both.
Hiura was enmeshed in a 0 for 13 slide and was batting ninth. And 37 of his 42 career home runs have come against right-handers, but he added Detwiler to his left-handed victims. It was the first run given up by Detwiler since joining the Reds and it pasted a defeat onto his record.
Lauer pitched through one out in the eighth and manager Craig Counsell brought in strikeout maven Devin Williams.
Jonathan India led the eighth off with a single, extending his hitting streak to eight games. But he was caught trying to steal second and Williams struck out Drury and Pham.
The Reds thought they caught a break when the nearly perfect closer, Josh Hader, was on paternity leave.
That forced Counsell to go with former Reds No. 1 draft pick Brad Boxberger, usually the seventh-inning pitcher ahead of Williiams (the eighth) and Hader (the ninth).
Kyle Farmer singled up the middle, the potential tying run, to begin the ninth. TJ Friedl replaced him as a pinch-runner. Max Schrock, fresh off the injured list, pinch-hit and popped to third.
Nick Senzel walked on four pitches, moving the potential tying run to second and the game-winner on first.
Mike Moustakas grounded behind the first base bag and Tellez made an excellent stop and zinged a bullet to second base for a force.
That put runners on third and first. . .and Bell sent Joey Votto up to pinch-hit. Votto was on the original lineup card but was a game-time scratch.
Votto took two pitches for balls, swung and missed and fouled one into the seats to go 2-and-2. He then flied to the warning track in right center and that was that.