The Brewers won the final two games of the series after the D-backs won the opener to snap their 17-game losing streak. Arizona has now lost 19 of its last 20 games and 33 of 36.
Smith allowed just one run on four hits over six innings, but walked away with a loss to drop his record to 2-3.
"Caleb threw the ball extremely well," D-backs manager Torey Lovullo said. "He gives us six unbelievable endings and, you know, you hold the team to one run over six innings, you're expecting to win a baseball game. But unfortunately he had somebody on the other side matching up his zeroes."
That somebody was right-hander Brandon Woodruff, who held the D-backs to one run on three hits over seven innings.
The lone run off Woodruff came on Christian Walker's seventh-inning homer.
"We knew what we were going to get," Walker said. "We knew what we were up against. It was just tough to get a rally going."
The one pitch that Smith would like to have back was one that he threw to Woodruff in the fifth with a runner on third. The fastball caught too much of the plate and Woodruff knocked it into center to plate the game's first run.
"It was just a heater middle-middle," Smith said. "I think that's what I'm most frustrated with about the outing was giving up a hit to him -- and not only hit but the go-ahead RBI. So I'm pretty upset about that."
There also appeared to be some upset in the D-backs dugout following that top of the fifth as the television broadcast caught outfielder David Peralta animatedly talking to his teammates in the dugout.
The D-backs caught a break -- if not the ball itself -- when the umpires called the infield fly rule, resulting in Taylor being ruled out.
Still, it was a sloppy play in a season of sloppy plays.
"To be honest, emotions are high," Walker said. "We take this stuff really seriously. ... We pour blood, sweat and tears into this every single day. And we expect a lot out of ourselves. I know me personally, I feel a lot of frustration towards myself. I want to be able to help the team more and that frustration and emotion comes out in all different ways. It's a lot of passion. And it's why Dave's a leader on this team."
Walker wasn't near Peralta at the time of the incident, nor was Lovullo.
"I trust David and I've asked him to be that kind of leader, and if he sees something he doesn't like to come in and make sure that everybody knows how he feels about particular plays," Lovullo said. "And, look, we were sloppy [with] a couple of errors in it; we've got to be better. You've got to be better than that."