The Yankees rookie rocketed a shot to straightaway center that seemed bound for a two-run homer.
The sold-out crowd gave a huge cheer, and another after Matt Holliday struck out to end the inning.
"It was special", Bradley said.
Price (5-2) allowed seven hits, walked none and struck out eight in eight innings. Craig Kimbrel, who blew a save chance for the third time this season on Saturday in Boston's 16-inning loss, pitched the ninth for his 24th save. The Yankees are the last team in the majors to be shut out.
"It's been a long, long time", manager Joe Girardi said.
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"I just hit it to the wrong part of the park and the wrong center fielder", Judge said. "They got one off our closer, and we got one off their closer".
Betts' two-out homer in the third inning snapped a stretch of 24 consecutive scoreless innings for Boston. He made it 3-0 when he singled to lead off the sixth, took second on an error by second baseman Starlin Castro, third on a groundout and scored on Dustin Pedroia's second hit of the game.
In the makeup of an April 25 postponement, the Red Sox hit just four balls out of the infield and extended their scoreless streak to 22 innings.
Rick Porcello (4-12) gave up three runs - one earned - and nine hits in six innings.
Judge, who leads the majors in homers with 30 but hasn't gone deep since winning the Home Run Derby on Monday at Marlins Park, sent a David Price pitch to the deepest part of the park. "He came up big". The veteran, however, kept the Red Sox off the board and improved to 8-3 with a 3.54 ERA.
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