The win snapped Arizona's three-game losing streak, while Colorado has now dropped five straight.
Anderson won for the fourth time in as many starts since being promoted from Double-A Mobile in early May. Anderson allowed four hits and two walks (one intentional) while striking out three.
The lone blemish for Anderson came in the fifth when DJ LeMahieu led off with a home run to left.
"He's been throwing the ball good ever since he's been up here," D-backs manager Kirk Gibson said. "He gave us six strong innings. He mixed it up. LeMahieu sat on a breaking ball. But other than that, just flawless, didn't walk a lot of guys, kept the balls in play."
Though he was in Double-A Mobile this year when he was called up, Anderson spent last season with Triple-A Reno, which plays in the Pacific Coast League, and that helped him prepare for Tuesday.
The PCL, much like Coors Field, is not known to be kind to pitchers, with several stadiums like Reno and Colorado Spring located at high elevations where the ball carries off the bat and breaking balls tend to have less movement.
"I just tried to stick to my game plan and pitch off that," Anderson said. "If you try to pitch to the park, sometimes you're not very successful. If you stick to your game plan, usually you can be successful. Being in Reno last year was definitely a tester for me. That ball flies there. If you give up one run, you've got to limit it to that. Being able to limit the damage is huge."
What also helped Anderson was that his best pitch is a changeup, which translates to altitude better than a breaking ball. In fact, one of the few breaking balls he threw Tuesday was the hanger that LeMahieu hit for the homer.
Anderson benefited by a couple of fortunate breaks in the game's first two innings.
In the first, Charlie Blackmon did not see that Justin Morneau's line drive got past a diving Chris Owings at short so he was forced out at second on what should have been a single.
Then in the second the Rockies scored a run when Wilin Rosario was ruled safe at home by umpire Jerry Layne. However, the play was overturned when Gibson used his manager's challenge, wiping a run off the board for Colorado.
By the time LeMahieu hit his homer, the D-backs had the lead thanks to a three-run outburst in the fourth inning against Jorge De La Rosa.
De La Rosa (6-4) walked a tightrope of trouble in the first three innings as the D-backs had at least two runners on base in each of the frames, but were unable to deliver a run-scoring hit.
That changed in a hurry in the fourth when Nick Evans and Owings led off with back-to-back homers.
Owings was 3-for-4 and finished a single shy of hitting for the cycle. After being named the National League Rookie of the Month for April, Owings hit a little bit of a cold spell the last couple weeks, but lately has shown signs of snapping out of it.
"Would I say it's exactly the same feeling?" he said referring to the month of April. "Am I ever going to feel that same feeling again? Who knows. But you're always searching for it and tonight was a good start."
The blast for Evans was his first big league homer since Sept. 3, 2011, while it was the fourth of the year for Owings.
"Fastball away," Evans said. "I just tried to hit it hard and it carried out of there. It definitely feels like it's been a while. It was nice."
The D-backs managed to manufacture a run later in the frame thanks in part to a De La Rosa wild pitch to grab the 3-0 lead.
Owings helped the D-backs tack on a run in the eighth when his double scored David Peralta.
That would prove to be important as Troy Tulowitzki led off the ninth with a homer off closer Addison Reed.
Corey Dickerson followed the homer with a single to bring the tying run to the plate, but Reed set down the next three hitters to wrap up the win.