Greene gave up his 21st and 22nd home runs, the most in MLB, but was only down a run when he left in the sixth. Reiver Sanmartin came on to give up a two-run double.
Luis Cessa followed Sanmartin and gave up another run as the bullpen continued its parade of ineptitude.
Greene (3-10, 6.01) was in dire straits in the first inning when the Mets loaded the bases on a hit by pitch, a walk and a single. Greene escaped by coaxing a first-pitch pop-up from Eduardo Escobar.
He was not so fortunate in the third. He hit Mark Canha with a pitch, No. 9 hitter Tomas Nido singled and leadoff hitter Brandon Nimmo drilled a three-run homer.
The Reds tied it in identical fashion in their third — hit by pitch, single, home run. Matt Reynolds was hit, Aramis Garcia singled and former Met Brandon Drury drilled his 17th home run, only the fourth homer given up by Mets starter Taijuan Walker this season.
Walker pitched six innings and gave up the three-run homer to Drury, four hits, one walk and struck out nine.
The Mets barged back in front on another home run off Greene, a two-out solo shot in the fifth by Francisco Lindor.
Greene gave up a single to Jeff McNeil and a one-out double to Canha. That was the end for Greene. Sanmartin replaced him and No. 8 hitter Dominic Smith lined a two-run double over center fielder Nick Senzel’s head to make it 6-3.
Greene pitched 5 1/3 innings and gave up six runs (two of his runners scored when Sanmartin gave up the two-run double), seven hits, walked one and struck out five.
Cessa trudged to the mound in the seventh and gave up a leadoff single to Lindor and a two-out run-scoring single to Escobar and it was 7-3.
The Reds retrieved one run in the bottom of the seventh against relief pitcher Drew Smith. Senzel hit his second home run of the season with one out. Matt Reynolds drove a double off the top of the wall but was stranded when pinch-hitter Max Schrock lined to right and Jonathan India struck out for the third time, leaving it 7-4.
Cincinnati produced only seven hits and was 1 for 6 with runners in scoring position.
The second game of the series Tuesday matches two pitchers coming off injuries and rehab assignments — Cincinnati’s Nick Lodolo against New York’s Max Scherzer.