Bradley, who was unscored upon in his first 6 1/3 innings of the spring, found himself in trouble early as he walked Endy Chavez to lead off the game.
Chavez stole second and moved to third when the throw got away and then scored on Robinson Cano's groundout.
In the second, Bradley once again walked the leadoff batter, Nick Franklin, and after consecutive singles by Dustin Ackley and Michael Saunders, the Mariners led, 2-0.
Ackley has been hot all spring, hitting .452 (14-for-31) with nine runs and nine RBIs.
Later in the second, back-to-back RBI singles by Willie Bloomquist and Cano put the Mariners up, 4-0.
Bradley wound up allowing four runs on five hits and two walks over two innings.
"Seattle swung the bats very good," D-backs manager Kirk Gibson said. "They're a good team. He got behind in the count, and he'd throw fastballs and they were whacking it."
Seattle starter Blake Beavan, on the other hand, was very impressive.
The right-hander, who is battling for a rotation spot, helped his chances by tossing 4 2/3 innings while allowing just one run and five hits. He did not walk a batter and struck out four.
"I like where I am right now compared to last year," said Beavan, who went 0-2 with a 6.12 ERA in 12 outings last season while dealing with tendinitis in his left knee. "I think it's a night-and-day difference. And I feel like I'm showing that, so I'm just going to keep doing what I'm doing."
Mariners manager Lloyd McClendon has been impressed by the 2007 Rangers first-round Draft pick.
"I thought he threw the ball extremely well. He got his pitch count up to 80 and was pretty impressive," McClendon said. "For the most part, he hit his spots. He had one bad 0-2 pitch, but other than that I was very, very pleased."
The lone run off Beavan came in the fourth, when Eric Chavez led off with a double and eventually scored on a groundout by A.J. Pollock.
Logan Morrison, who is competing for the Mariners' starting DH spot, hit his first homer of the spring in the fifth inning. The ball landed halfway up the grass berm beyond the wall in right-center.
Up next: Felix Hernandez makes his third Cactus League start Friday at 6:40 p.m. PT on Gameday Audio as the Mariners face the Rockies at Salt River Fields. Hernandez gave up four hits and four runs in 2 1/3 innings in his last outing against the Rangers, but said he feels strong as he prepares for his club-record seventh Opening Day start. Erasmo Ramirez is scheduled to pitch in relief and will get an extended outing as he's competing for a starting-rotation berth. Right fielder Corey Hart is expected to return to the lineup after being sidelined since Saturday with a stiff lower back.