After three scoreless innings, Greene allowed three runs to open the fourth – a Rowdy Tellez single, an Andrew McCutchen single and a three-run home run for Hunter Renfroe.
And the Brewers made it 4-0 in the fourth when Willy Adames hit the right spots on the field with a home run.
The Reds used the same modus operandi to get back into the game against Milwaukee southpaw and Kent State product Eric Lauer.
Mike Moustakas fired a right-out single in the fifth and Albert Almora Jr. reached the left-field seats for a home run that put the Reds up 4-2.
It took the first two pitches of the sixth inning for the Reds to tie, 4-4. Brandon Drury and Tommy Pham hit back-to-back homers on the first pitch.
Lauer is sensitive to the long ball. He allowed 16 home runs this season and was touched for three home runs in a game three times.
Greene, however, 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 an expected left-handed streak, Bell went to left-hander Ross Detwiler with one out in the seventh.
He retired southpaw Jace Peterson, but southpaw Keston Hiura left the field with a home run for a 5-4 Brewers lead that stood the test of time.
The Reds put two runners on base with one out in the ninth but blocked them both.
Hiura was entangled in an 0-for-13 slide and was ninth at bat. And 37 of his 42 career home runs have come against right-handers, but he added Detwiler to his left-handed victims. It was Detwiler’s first run since joining the Reds and it stuck a loss to his record.
Lauer had an out in the eighth and manager Craig Counsell brought in batting maven Devin Williams.
Jonathan India led the eighth with a single, extending his hitting streak to eight games. But he was caught stealing the second and Williams knocked out Drury and Pham.
The Reds thought they had taken a break when near-perfect relative 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 Williams (the eighth) and Hader (the ninth).
Kyle Farmer selected the middle, potentially even run, to start ninth. TJ Friedl replaced him as the pinch racer. Max Schrock, fresh off the injured list, pinch punched and moved up to third.
Nick Senzel walked four pitches, moving the potential link-up run to second and the match-winner to first.
Mike Moustakas crashed behind the first base sack and Tellez made a great save and sent a ball to second base for some force.
This placed the runners third and first. . .and Bell sent Joey Votto to pinch. Votto was on the original composition card but was a scratch during play.
Votto took two pitches for the balls, swung and missed and fouled the seats to go 2 and 2. He then flew to the warning lane at center right and that was it.