Bengie Molina's two-run home run gave San Francisco a 3-1 lead in the third before Davis walked Pablo Sandoval in an 11-pitch at-bat and gave up a two-out single to left field to Kevin Frandsen.
Andres Torres followed with a high drive to the warning track in center field that Young could not track down after a long run. Both runners scored and Torres ended up at third.
"It looked like he got turned around. He ran to a spot, which is what you teach your outfielders to do," manager A. J. Hinch said. "But he ran to the spot a little further off than where the ball would be. The ball caught him off guard a little.
"It was a difficult play, but, again, it is one of those plays that you want to be made, because it changes the course of the game. Whether it is an error or a base hit is debatable; the outcome was not what we wanted."
Young said he was playing Torres, a .217 hitter with one home run in limited time this season, fairly shallow and would do it again.
"It was a tough play," Young said. "Back to the wall, and I'm trying to get there. He's not a power guy, so I'm playing him according to where I should play him. Every hitter I play differently. That's why I had him positioned there, and I would be in the same spot tomorrow."
The D-backs got three runs back in the fifth inning off Barry Zito (3-6) to cut the deficit to 5-4 but did not get a hit in the final four innings, as Giants relievers retired the last 10.
Shortstop Stephen Drew singled in a run in the first to extend his hitting streak to 14 games, and his two-run double in the fifth inning capped a three-run inning.
Felipe Lopez singled with one out and took third on Ryan Roberts' double off the left-field fence before Justin Upton lined a single to right to score Lopez and bring up Drew.
Mark Reynolds followed Drew's double with an infield single into the hole at shortstop before Young fouled out to third base.
After Chris Snyder walked to load the bases, pinch-hitter Miguel Montero hit a long foul down the right-field line that Randy Winn caught while leaping into a protective pad about six feet short of the foul pole.
Roberts walked with two outs in the sixth, the D-backs' only other baserunner.
"We need to get some leads and tack on some runs," Hinch said.
The D-backs have committed 51 errors, third in the National League, behind only Florida's 52 and Washington's 56.
Davis gave up six hits and three walks in four innings, when he had to be lifted after hitting 100 pitches.
"I felt I got lucky," Davis said. "I didn't feel like I had anything out there. I got behind a lot of hitters. I was ball one, ball two. Errors are going to happen. I'm the one that got it started. You can't expect everyone to be sharp when I'm keeping them out there so long."
With two outs in the top of the ninth inning, a young female fan jumped onto the field from the left-center-field stands and ran to center, where she stopped to hug Young. The lady was escorted off the field by security.
"It was fun. She was fine. She didn't do anything out of line. She gave me a hug, and I gave her a little hug back," Young said.
Young was questioned by the authorities after the game and said he does not want to press charges.