Each of the A's last seven losses have been by one run, the longest such streak in Oakland history.
The latest, manager Bob Melvin said repeatedly, was "frustrating, obviously."
Balfour, who had not allowed a run over his past 10 games, was tagged with four runs in the eighth inning, when he allowed a one-out single to Elvis Andrus before walking Josh Hamilton and, after getting Adrian Beltre to fly out, Michael Young. Another free pass to Nelson Cruz brought in a run, at which point Ryan Cook entered and surrendered a three-run triple to Craig Gentry that found its way over Yoenis Cespedes in left-center.
After the game, Cespedes was asked what type of read he got on the ball, as he appeared to take a bad route. He said, "I didn't do it the right way, but even if I read the ball well, I had no chance to catch it."
Later, he added, "Maybe if I played left field all the time in my life before, maybe that's not going to happen."
"I knew I hit it pretty good, but off the bat I thought he was going to catch it," Gentry said. "Luckily I got a good pitch to hit and I was able to get the barrel of the bat on it."
It proved to be enough damage for a second straight loss in Texas for the A's, who couldn't hand Griffin a win in his second big league start, after he retired 18 of 22 batters faced, with help from four strikeouts -- including three straight to start the game -- and 10 ground-ball outs.
"I thought he threw well, really well," Balfour said. "That's what [stinks], too. You want to see a guy get a win out there for that kind of outing. It's frustrating, just a tough day.
"I didn't throw strikes and I walked guys, and I leave myself in a bad situation. That's the way it is. Didn't get the job done. Anytime you lose it's frustrating, but especially like that."
The 24-year-old Griffin, called up Saturday to replace the oft-injured Brandon McCarthy in a battered A's rotation, has now compiled 11 consecutive shutout innings, even after finding trouble in the second inning on Friday.
With one out, Griffin walked Young and allowed a base hit to Cruz, only to issue another free pass to David Murphy to load the bases for Mike Napoli. The righty left Napoli frozen on strikes, and Leonys Martin lined out to first to put the frame to rest.
"I didn't have my fastball command that I usually do, but I just tried to battle through it," Griffin said. "They did a good job of fouling off pitches and getting deep in counts, but I just tried to keep us in the game. That's what you try to go out there and do every time. I got them to swing at pitches I needed them to swing at most of the time, and it just worked out in my favor."
The A's, meanwhile, posted a pair of runs in eight innings off Rangers lefty Matt Harrison, an All-Star candidate who earned his 11th win, good for a tie atop the American League.
In the fourth, Jemile Weeks tallied his second hit of the night, a leadoff triple to left-center, and quickly scored on Cespedes' base hit. Coco Crisp, who appears to be back on track at the plate since returning to the leadoff spot, gave the A's their second run in the fifth courtesy of an RBI single.
Chris Carter, making his A's season debut, homered in the ninth off Tanner Scheppers, to slim Texas' lead, but it wasn't enough to prevent Melvin and Co. from staring down another close defeat.
"We're either battling back or losing it at the end, one way or another," he said.