Reddick, who's been hot so far this spring, gave the A's an early lead when he hit his first homer of the spring, a three-run blast to right-center field, in the bottom of the first. The A's right fielder then went the other way in his next at-bat and doubled to left-center field.
The homer put the Giants in an early deficit, but came just two batters after the club weathered an early scare.
On Johnny Cueto's first pitch of the night, Burns hit a line drive right back up the middle that ricocheted off Cueto's forehead.
Cueto stayed in the game and threw three innings. The right-hander struck out two and gave up three earned runs on five hits and a walk, but went back to Scottsdale Stadium after his outing to get checked by team doctors for a concussion.
"I was hoping it was what it was, more of a glancing blow," Giants manager Bruce Bochy said. "Still it caught him pretty good."
Meanwhile, Sonny Gray mixed in all his pitches and looked the part of an Opening Day starter.
"We kind of treated it like a start, like a regular-season start," Gray said. "It was a pretty good atmosphere; it always is when you play those guys, no matter where we play. We threw everything, we got some really, really good work in."
As Reddick was providing the A's with offense, Burns, who also doubled and drove in two runs, was doing his best to stifle the Giants' bats.
Burns saved a run and took a hit away from San Francisco's Jarrett Parker with a diving catch to end the top of the first. The outfielder then struck again in the third when he made another diving catch, this time taking a single away from Ehire Adrianza.
"Unbelievable today," Melvin said. "This was the best defense we've played all spring. I have six pluses on my card."
But Burns couldn't get to every ball, and when he was unable to make the spectacular catches, the Giants were able to scratch some runs across.
Matt Duffy led off the fourth inning with a homer to left-center and Gorkys Hernandez drove in a run with a triple to straightaway center, just out of Burns' reach.
That was all the offense the Giants could muster off Gray. The right-hander retired nine in a row at one point and finished his outing after 4 2/3 innings, allowing four hits and striking out four.
Giants Up Next: Following their lone scheduled off-day of the spring Tuesday, the Giants will resume the Cactus League campaign with a 1:10 p.m. PT contest Wednesday against Seattle in Peoria. Jeff Samardzija will start for San Francisco against fellow right-hander Hisashi Iwakuma, live on MLB.TV and via an exclusive video webcast.
A's Up Next: After a pair of shaky outings, Rich Hill looks to hone his command against the Rockies in Scottsdale on Tuesday at 1:10 p.m. PT, live via an exclusive audio webcast. The veteran left-hander gave up seven runs in 2 1/3 innings in his last outing and has walked three batters in each of his first two starts. Sean Manaea, the A's No. 2 prospect according to MLBPipeline.com, and Fernando Rodriguez will also pitch.
William Boor is a reporter for MLB.com. Follow him on Twitter at @wboor. This story was not subject to the approval of Major League Baseball or its clubs.