The Isotopes scored 11 runs in the top of the first before barely holding on.
Brady started in place of veteran Joe Martinez, who retired on Saturday after struggling this season.
"It kind of threw us for a loop early," Salt Lake manager Keith Johnson said. "But I wish [Martinez] nothing but the best. Those things happen in baseball."
Beyond the first inning, Johnson was thrilled by the effort his players gave in nearly overcoming the huge early deficit.
"They did a good job," he said. "They played for 27 outs. Even though we gave up that big first inning, we've still got 27 outs to try and score 11 runs and we're going to do everything we can.
"We're going to take the extra base. We're going to bunt. We're going to do everything to try and get guys on [base]. Our guys did a good job. They just came up a little short at the end."
Grant Green went 3-for-4 with three RBIs for the Bees, who had a two-game winning streak snapped.
"The mindset, whether we're up 11 or down 11, is pretty much the same," Green said. "Give quality [at-bats and] keep battling. ... We may not win the game, but we're going to put up a fight and screw the other guy's ERA."
Brady is a converted infielder who started this season at Arkansas. He also reached Double-A in the Miami Marlins' organization last year, when he posted an ERA of 1.53 in 49 relief appearances at Jacksonville.
In the top of the first inning, however, Brady retired only one of Albuquerque's first eight batters and allowed seven earned runs.
"He was behind the eight-ball a little bit," Johnson said. "He hadn't pitched in five days and it was his first start of any kind since he's been a pro. ... I have no doubts the next time we put him out there he's going to take the ball and run with it."
The Isotopes eventually bunched nine hits and two walks off Brady and reliever Jeremy Berg. Trayvon Robinson drilled a three-run homer and Clint Robinson added a two-run blast during the explosive start that doomed the Bees.
Still, Salt Lake fought back, partly because its bullpen allowed only two runs and scattered eight hits over the final eight innings.
In the bottom of the ninth, Salt Lake trimmed a game-long deficit to one run.
Roberto Lopez led off with a double into the left-field corner. He went to third on Efren Navarro's infield out and scored on Shawn O'Malley's grounder. But with two out and nobody on, Colt Hynes preserved Albuquerque's win by getting Taylor Lindsey on a come-back grounder to the mound.
Isotopes 13, Bees 12
• Albuquerque scores 11 runs in the first inning during a PCL victory over Salt Lake.
• Third baseman Grant Green goes 3-for-4 with three RBIs to lead the Bees' 14-hit attack.
• This series resumes Monday night, when Matt Shoemaker will pitch for Salt Lake.