Bradley made a spectacular catch to rob Judge of a two-run homer in center field in the eighth inning, preserving Price's gem as the Red Sox earned a doubleheader split with a 3-0 victory at Fenway Park.
"It's one of those balls you try to time up," Bradley said of his incredible catch. "It was in the air. Those are the type of plays where they rarely happen. A lot of different things (have) got to go the right way."
Bradley timed the catch perfectly from the crack of the bat, tracked it to the wall of the triangle in center and reached into the Red Sox's dugout for the snag.
Price celebrated Bradley's catch emphatically, and the Fenway crowd roared and gave the center fielder a lengthy standing ovation.
"(The reaction) actually made the hair stand up on my arm," said Red Sox right fielder Mookie Betts, who provided the bulk of Boston's offense with a two-run homer.
Boston (52-41) salvaged a split of its four-game series with New York (47-43) after losing the first game of the doubleheader 3-0.
With the win, the first-place Red Sox moved three games ahead of the Tampa Bay Rays and 3 1/2 ahead of the Yankees in the American League East.
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.
"On a night when we needed a bounce-back win, it was a big performance by David tonight," Red Sox manager John Farrell said.
The Yankees were shut out for the first time this year. New York has not won any of its past nine series, its longest such streak since 1991.
"Tough place to come and play, and we ended up 2-2," Yankees manager Joe Girardi said after a series that saw 1,377 pitches thrown in a combined time of 15 hours and 35 minutes.
Betts finished with three hits, including the homer, his team-leading 17th. Dustin Pedroia added two hits and drove in a run for the Red Sox, tying Nomar Garciaparra for 16th place on Fenway Park's all-time RBIs list with 380.
Masahiro Tanaka (7-9) took the loss after pitching 7 2/3 innings, giving up three runs, eight hits and no walks while striking out nine.
"I thought he made one bad pitch," Girardi said of Tanaka. "One slider that really didn't do much. Besides that, he was really good."
No Yankees batter had more than one hit, and Judge went 0-for-4 with two strikeouts.
Betts' two-out homer in the third inning snapped a stretch of 24 consecutive scoreless innings for Boston.
The blast came on a 1-1 slider from Tanaka and sailed an estimated 410 feet over the Green Monster in left field.
Garrett Cooper, who made his major league debut for the Yankees on Friday, doubled in the fifth for his first major league hit.
Pedroia made it 3-0 with his RBI single in the sixth.
Judge nearly brought New York within a run with his deep fly ball two innings later.
"I just hit it to the wrong part of the park and the wrong center fielder," Judge said. "Jackie's been making plays like that for a long time."
In New York's victory in the opener, the Yankees got a solo homer from Gregorius, a sacrifice fly from Ji-Man Choi and an RBI single from Ronald Torreyes.
CC Sabathia (8-3) tossed six scoreless innings to outduel Boston starter Rick Porcello (4-12).
NOTES: Yankees RHP Bryan Mitchell and Red Sox INF/OF Brock Holt were the 26th men for Sunday's doubleheader. Boston returned INF Tzu-Wei Lin and RHP Austin Maddox to Triple-A Pawtucket after the game. New York sent RHP Domingo German to Triple-A Scranton/Wilkes-Barre. ... Yankees SS Didi Gregorius' fifth-inning homer to right field in Game 1 traveled 295 feet, per Statcast. However, the shortest home run distance to right at Fenway Park is listed as 302 feet. It was the shortest non-inside-the-park homer Statcast has tracked in the past three seasons. ... Red Sox manager John Farrell said the team will follow through on its formal protest of Saturday's game, despite New York DH Matt Holliday's interfering slide back to first base in the 11th inning not having an impact on the final score. "If it goes unaddressed ... who's to say (you) can't instruct runners to do the same going forward," Farrell said. ... Mitchell (1-1, 5.06 ERA) will face Minnesota Twins LHP Adalberto Mejia (4-4, 4.43 ERA), while Red Sox LHP Eduardo Rodriguez (4-2, 3.54 ERA) opposes Toronto Blue Jays RHP Marcus Stroman (9-5, 3.28 ERA) on Monday.