With the loss, the D-backs missed sweeping the three-game set.
D-backs left-hander Oliver Perez started the 10th in dominant fashion, striking out the first two batters he faced, and he jumped ahead of Dickerson 0-1 before hanging a slider that Dickerson hit into the pool area beyond the wall in right-center.
"One pitch can change everything, and that was the pitch," Perez said. "I was going to throw it down in the zone and I just slipped it right in the middle. That was a mistake and he took advantage."
After using a full leg kick on the first pitch to Dickerson, he changed things on the second one.
"I went up there looking for something out over the plate," Dickerson said. "He slide-stepped that second pitch and I just used my hands as a reaction. I thought it was a line drive. I thought it was going to hit the wall but it went out."
The homer came on the heels of a golden opportunity for the D-backs to win the game in the bottom of the ninth as they loaded the bases with one out.
David Peralta, though, hit a grounder to short and the Rockies were able to force the runner at the plate, and then Mark Trumbo grounded out to end the threat.
With runners on first and second during that inning, Aaron Hill hit a bullet to left, but the ball was hit so hard that Dickerson was able to get it back into the infield quickly so the runner had to hold at third.
That was one of just two hits the D-backs had on the day with runners in scoring position.
"We had pressure all over," D-backs manager Kirk Gibson said. "We had a lot of opportunities. Other than that, we had an opportunity to win that game in the ninth inning, and we let it slip away. That's a game we should win."
The D-backs grabbed an early lead against Franklin Morales thanks to a two-run first-inning double by Trumbo.
Wade Miley, though, has struggled this year to shut the opponent down in the inning after being given a lead and he quickly let the Rockies right back in this one.
After Drew Stubbs drew a walk to lead off the second, Ben Paulsen hit his first career big league homer to tie the game at 2.
"The pitch to Paulsen, tried to steal a strike and he stayed on it for a two-run," Miley said. "That kind of hurt. That was a big part of the game right there. We'd just scored two runs but to give it right back kind of hurt."
Sloppy defense by the Rockies allowed the D-backs to get the lead back in the fourth.
Alfredo Marte and Jake Lamb led off the inning with singles and an error by Morales loaded the bases with nobody out.
Morales then got Didi Gregorius to pop out and it looked like he might escape the inning unscathed when Miley hit a tapper back to the mound. Morales flipped to catcher Wilin Rosario for the force at home, but Rosario's throw to first to complete the double play sailed down the line and a run scored to give Arizona a 3-2 lead.
The home run has been trouble for Miley, who has allowed the second most in the National League, and once again it was the long ball that allowed the Rockies to tie the game when Rosario hit a one-out homer in the sixth.
"Well, Rosario, he pitched pretty good with his offspeed stuff," Gibson said of Miley. "He tried to sneak a fastball by him, he hit it out."