The Athletics' lone hit was a groundball single to right field by Shannon Stewart with two outs in the bottom of the ninth inning.
"Don't give up a hit," Schilling told New England Sports Network when asked what he was thinking when Stewart came to the plate. "It was 1-0 the whole day, so that alleviated thinking about anything else. I just wanted to finish that game."
The ace right-hander allowed four earned runs and nine hits in his previous outing last Saturday against the New York Yankees.
Despite more than 3,000 strikeouts, 200 career wins, three 20-win seasons and a World Series championship ring in both leagues, the 40-year-old Schilling has never pitched a no-hitter or been named a Cy Young winner.
He would have been the third oldest pitcher behind Nolan Ryan and Cy Young to throw a no-hitter.