The D-backs led right up until pinch-hitter Bengie Molina hit a three-run homer off closer Chad Qualls with two outs in the eighth inning.
"Not just for me, obviously, frustrating for the whole team," Davis said. "One pitch away, you know."
Davis cruised through 7 2/3 innings before allowing a walk to Edgar Renteria and a bloop hit to Randy Winn.
"He put a good at-bat on me," Davis said of Renteria. "I buried the cutter in on Winn, and he just happened to fist it out there to right field."
Arizona manager A.J. Hinch then went to Qualls, and San Francisco countered by hitting Molina for Ryan Garko. After jumping ahead of Molina 0-2, Qualls hung a slider that wound up in the bleachers.
"Obviously I felt like I had more in the tank to go ahead and get Garko, since he hadn't hit one out of the infield yet," Davis said. "Not to say they wouldn't have pinch-hit Bengie then, but I believe they had announced [Garko's] name already, so I thought I was going to face him there. What can you say? It didn't work out."
When asked if he wanted to stay in the game, Davis, who had thrown 98 pitches, said, "Do I ever want to come out of the game? It's obviously not my call. You've got to respect the call of the manager and deal with it. It's not always easy to deal with it."
"It's a frustrating loss," Hinch said. "I don't really care to get into it too much, but it's a frustrating loss for everybody -- certainly a starting pitcher that throws into the eighth inning and ends up with a no-decision. He put a lot of heart and soul into that game. But it is what it is."
Steve Gilbert is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.