The runners were off with the pitch on what Reds manager Bryan Price said was a double steal, so Carter stepped on the first-base bag to double-up Scott Schebler, before tossing the ball to shortstop Orlando Arcia covering second base to retire Jose Peraza for a sudden end to the Reds threat.
Mark it 3-6 on your scorecard.
"We got three outs on a play when Joey Votto was at bat. That was the biggest thing for me," Brewers manager Craig Counsell said. "It was pretty incredible. That had a chance to be a big inning, for sure."
"To get out of that with nothing was awesome," Davies said.
It marked the first time a Reds batter hit into a triple play since Edwin Encarnacion's 6-4-3 lineout on April 12, 2009.
"We scored five runs at Chicago in three days," said Reds manager Bryan Price, explaining why he put the runners in motion. "At some point in time, we'll have to break out and be a stronger offense club to try and get some distance between ourselves and the other clubs. ...
"It's like any move you make. If it doesn't work out, you look like you don't know what you're doing. If it does work out, you look like a genius. There's a lot of risk-reward in this position, as you guys can imagine. That was one where you go, 'Oh geez.'"
Schebler said he saw Votto's line drive leave the bat and figured there were two possible outcomes: Foul ball, or double down the line. Then he heard the crowd react, and realized there was a third option.
"I did the math as I was walking back" to the dugout, he said. "You don't see that very often. It stinks to be on that end of it, especially with a hard-hit ball like that."
Carter had more experience in the matter. He's started just 377 games at first base in the Majors but has already been part of four triple plays, including both of the Brewers' this season. He also contributed to one in 2012 with Oakland, and in 2015 (with Jonathan Villar) in Houston.
As the runners broke from their bases with Davies' pitch, Carter said, "I knew there was a potential for it. I ended up catching it and touching first and then [going to] second. My first thought was to touch first because I knew they both ran."
"We talk to Chris a lot about his defense," Counsell said. "I think when you hit that many home runs, your defense gets overlooked a lot. But it really is important."
The Brewers' other triple play this season went 5-4-3 against the Marlins on April 29 at Miller Park. Miami's Marcell Ozuna hit a sharp grounder to Brewers third baseman Aaron Hill, who relayed to second baseman Yadiel Rivera, who relayed to Carter.