Manaea struck out 10, walked two and threw 108 pitches to finish off Oakland's first no-no since Dallas Braden tossed a perfect game against Tampa Bay in 2010.
Manaea got Hanley Ramirez to ground out to complete the first no-hitter versus the Red Sox since Seattle's Chris Bosio did it on April 22, 1993.
"I just telling myself, keep everything the same, and not let anything get too big for me," Manaea said.
Boston looked as if it had a hit with two outs in the sixth at the Oakland Coliseum. Andrew Benintendi hit a grounder to the right side, tried to dodge a tag by first baseman Matt Olson and was called safe.
After the umpires conferred, Benintendi was ruled out for going wide of the baseline. Benintendi ran onto the grass in foul territory to get around Olson.
"Do I agree with it? No," Red Sox manager Alex Cora said. "It is what it is."
Sandy Leon reached in the Red Sox fifth when A's shortstop Marcus Semien ranged into shallow center field and dropped a popup trying to make an over-the-shoulder catch. The play was scored as an error.
Manaea said he figured it was a hit.
"I didn't even think about it until I looked up in the seventh or eighth, and I was like, 'Oh my God, why is there still a zero on there?'" Manaea said. "So after that, my adrenaline started pumping a little bit again, and I really wanted to finish this thing out."
Manaea (3-2) had been battered by Boston in three previous starts, going 1-2 with a 13.50 ERA.
But the 26-year-old cooled off a Red Sox team that had won eight in a row and 17 of 18. Boston began the game with a major league-best .293 batting average.
Manaea retired 14 straight batters after walking Mookie Betts leading off the game.
A lot of Red Sox fans were in the announced crowd of 25,746, and many rose to their feet when Manaea took the mound in the ninth.
Betts hit a ball to the warning track with one out in the ninth. After Benintendi walked with two outs on a full count, Ramirez hit a grounder to Semien for an easy force play at second.
"My heart was beating out of my chest and I was trying to do everything I could (to stay calm)," Manaea said before he was interrupted by teammates who smeared celebratory shaving cream in his face, dumped water and Gatorade on his head and left him with a plastic bubble gum tub for a hat.
This was the first no-hitter in the majors since Miami's Edinson Volquez pitched the only one of the 2017 season against Arizona on June 3.
Chris Sale (1-1) struck out 10 and gave up three runs on six hits and one walk in seven innings.