Arizona led, 5-1, heading into the ninth when Gibson elected to go with left-hander Eury De la Rosa to start the inning. De la Rosa gave up a single and a walk while retiring just one batter before Gibson turned to closer Addison Reed.
After Reed got Todd Frazier to fly out, Devin Mesoraco made things interesting with a three-run homer to left to pull the Reds to within a run.
Reed, though, settled down and struck out Ryan Ludwick to end the game and record his 26th save.
"It's not really the way we drew it up, but nevertheless we had enough runs," Gibson said. "He had a lead and you're going to throw strikes. You weren't going to walk that guy. You're going to go after him. Mesoraco had a pretty good series against us. He wasn't the tying run. [Reed] got the tying run out."
That they did thanks to a two-run homer by Paul Goldschmidt in the sixth, a sac fly by Mark Trumbo in the eighth and a two-run blast by Didi Gregorius in the ninth.
"They were able to create that separation," Reds manager Bryan Price said. "We talk about separation all the time, and what they did is they created the separation and got the two-run homer and then the add-on run in the eighth and the two runs in the ninth. And that's just not a good place to start turning your offense on when you're down four in the ninth."
The D-backs' offense struggled to put anything together against Reds starter Alfredo Simon, who held Arizona to just two hits through the first five innings.
Finally Goldschmidt was able to break through with the two-run homer in the sixth.
"It was a fastball," Goldschmidt said. "He'd been doing a good job keeping it down and fortunately that one stayed up and I was able to hit it hard and get it out of there."
That Goldschmidt's homer gave the D-backs a lead was in part because of the way starter Wade Miley was pitching.
Miley (7-7) tossed 6 2/3 solid innings to win for the fourth time in his last five starts. He had been looking to bounce back after allowing six runs over five innings in his last start against the Phillies.
It wasn't exactly a thing of beauty, though, as Miley walked four and allowed seven hits.
"It was a battle," Miley said. "I was able to get through it. It got better as the game went on, but early on I was just kind of fighting, just didn't feel very good. They were hitting it at some people so we were able to get some outs and get out of there."
Miley was supported by outstanding defense from Gregorius at short and Gerardo Parra in right field. Trumbo also came up big in left field, throwing out Chris Heisey trying to stretch a single into a double.
Gregorius, who was traded by the Reds to Arizona in a three-team deal before last season, was 6-for-14 in the series, but said there was no added motivation in playing against the team that dealt him.
"For me, I just go out there and play the game," Gregorius said. "You're still competing. The main thing for me is trying to compete and have fun before or after the game. We're out here to play baseball. It doesn't matter who we play against. We always give 100 percent."