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 earlier Sunday afternoon.
With the win, the Red Sox moved three games ahead of the Tampa Bay Rays and 3 1/2 ahead of the Yankees atop the American League East.
Price (5-2) allowed seven hits, walked none and struck out eight. Craig Kimbrel, who blew his third save of the season in Saturday's 16-inning loss, pitched the ninth for his 24th save.
It was the first time the Yankees have been shut out this year.
Betts finished with three hits including the homer, his team-leading 17th. Dustin Pedroia also drove in a run for the Red Sox and tied 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, seven hits and no walks while striking out nine.
No Yankees batter had more than one hit, with Aaron Judge going 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 to center in the eighth, but Jackie Bradley Jr. reached up and made a spectacular catch to rob Judge of a two-run homer.
Didi Gregorius doubled with one out in the ninth and Kimbrel walked pinch hitter Brett Gardner after a fly out to bring up another pinch hitter in Chase Headley, who struck out to end the game.
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: New York RHP Bryan Mitchell and Boston INF/OF Brock Holt were the 26th men for Sunday's doubleheader. ... Yankees SS Didi Gregorius' fifth-inning homer to right field in Game 1 Sunday 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 last 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. ... The Yankees (starter TBA) will face Minnesota Twins LHP Adalberto Mejia (4-4, 4.43 ERA), and Red Sox LHP Eduardo Rodriguez (4-2, 3.54 ERA) opposes Toronto Blue Jays RHP Marcus Stroman (9-5, 3.28 ERA) on Monday.