Colorado's Troy Tulowitzki, Paulsen and Nolan Arenado each drove in first-inning runs, while Michael Cuddyer and Jason Pridie also had a hit in the first.
Griffin settled down after the opening frame and finished allowing only those three runs on six hits in three innings.
Rockies left-hander Brett Anderson threw three innings of one-run ball against his former team, surrendering three hits and striking out one.
"It was different," Anderson said of facing his former teammates. "I tried not to think about it before but then Coco [Crisp] gets in the box and tips his hat and you tip yours back. It's kind of surreal."
The A's lone run against their former teammate came in the first when Rockies center fielder Charlie Blackmon misjudged a liner off the bat of Oakland's Nate Freiman and attempted a diving catch. Blackmon missed, allowing Alberto Callaspo to score from first and giving Freiman a triple.
Anderson was dealt from Oakland to Colorado for fellow lefty Drew Pomeranz in December. Pomeranz gave up a run on two hits and a walk in 1 1/3 innings of relief.
The A's added a run in the fourth inning on John Jaso's RBI ground-rule double and another in the fifth on a Nick Punto single.
Up next: The Rockies travel to Surprise on Sunday to face the Royals at 12:05 p.m. MT. Right-hander Juan Nicasio looks to follow a sterling Cactus League debut.
Cash Kruth is a reporter for MLB.com. Follow him on Twitter at @cashkruth. This story was not subject to the approval of Major League Baseball or its clubs.