In dire need of a stopper, having watched the Red Sox walk all over them in their first five meetings this season, the A's got just that from Graveman, who worked around six hits to keep trading zeros with Red Sox starter Eduardo Rodriguez.
"Kendall's our ace right now," shortstop Marcus Semien said.
The A's were in danger of being no-hit by Rodriguez, before Semien reached first with two outs in the eighth on an infield single, courtesy of a Bob Melvin challenge and an overturned call.
Graveman was out of the game by then, with Ryan Dull picking him up with 1 2/3 clean innings, but there was still time to reward him.
Khris Davis almost felt it his duty to do so, and he did, doubling off Red Sox closer Craig Kimbrel to help bring Danny Valencia safely home from first base, with an assist from left fielder Brock Holt, who misplayed the ball.
"He kept me in the game," Davis said of Graveman. "Early in the game, I was a little frustrated with the umpire. [Graveman] showed some leadership, just letting me know he needs me out there, and it was a good feeling to have, and it showed up big. I give him credit."
Graveman, having hurriedly stepped up to the front of the rotation with Sonny Gray bothered by injuries and inconsistencies this season, has a 2.06 ERA over his last five starts, bringing his season ERA down to 3.81.
It was as high as 5.49 in June.
"To respond today and win a close game like that against a team that had really handed it to us, not only these two games here but over at their place, too," Melvin said, "we were really relying a lot on Kendall Graveman today to keep that game exactly how he did."
Graveman's performance included an escape act in the fourth inning, when he wiggled his way out of a bases-loaded jam with one out. He finished with seven strikeouts, fanning a formidable trio of David Ortiz, Mookie Betts and Hanley Ramirez in a row at one point.
The Red Sox had plated 27 runs in their first two games of the series and had outscored the A's, 67-19, in five games this season.
"I studied a lot this week," Graveman said. "Plan of attack was just to mix pitches, and I think that's what you have to do against a team that's hitting .280 as a group. To be able to get out of that inning with the bases loaded was huge. The way Rodriguez was throwing, you still have a chance. You're tied and at home, you think you can hit a walk-off at any moment, and that's what happened."
Jane Lee has covered the A's for MLB.com since 2010. Follow her on Twitter @JaneMLB.
This story was not subject to the approval of Major League Baseball or its clubs.