Manaea (12-10) gave up three hits and no earned runs in 6 2/3 innings.
"That's finishing on a strong note for a guy that was really grinding toward the end," A's manager Bob Melvin said. "He had a couple bouts this season with adversity and pitched through it."
It was the fifth straight road win for the Athletics and seventh consecutive loss for the Rangers. It's Texas' longest skid since an eight-game slide in 2015.
Bruce Maxwell took to a knee in the Oakland bullpen during the national anthem. The catcher was warming up Manaea, out of view from many fans and cameras. One fan near the Oakland dugout said "boo" in a normal voice, and a small number of fans booed when Maxwell batted in the second inning. Maxwell has been the only major league baseball player to take to a knee, mirroring protests by NFL players against racial injustice. This was Maxwell's first road game since he began his protest last weekend.
"I really didn't notice," Melvin said. "It's not something I was really focused on."
Manaea had been scratched from his previous scheduled start, at home against the Rangers on Saturday, because of a strained left upper back.
During that game, Manaea received a cortisone shot.
"Coming into today I didn't really notice it at all," he said.
Manaea won four of his last six decisions with an attitude adjustment after a three-game losing streak in August.
"I felt like I had to show that sad emotion, like I was defeated. It was tough for me to get out of those funks, especially last month in August."
Blake Treinen pitched a scoreless ninth for his 15th save in 20 opportunities.
With two out in the sixth, Healy looped a fly ball that dropped in front of center fielder Delino DeShields to drive home Jed Lowrie and Khris Davis. Matt Chapman also drove in two runs, with his 14th home run in the fifth and a single in the ninth.
Chapman's homer off losing pitcher Miguel Gonzalez (8-13) cleared the center field fence just above the outstretched glove of DeShields.
Oakland is 16-5 in the last 21 games. By beating fourth-place Texas, the A's retained hopes of avoiding their third straight last-place finish in the AL West. The Rangers have a two-game lead with three to play.
"There's guys on the bench that are sitting over there that have been the mainstay and the focal point of our offense."
Earlier in the day, the A's announced an extension of Melvin's contract through the 2019 season. His 536 wins with Oakland trail only Tony LaRussa (798) and Art Howe (600).
A STREAK ENDED
SS Marcus Semien's fifth-inning throwing error on a grounder by Willie Calhoun ended Oakland's streak of nine games without an error. The A's still lead the major leagues with 119 errors. Calhoun advanced to score the major league-high 83rd unearned run against Oakland.
A's: Rookie utility player Chad Pinder, who sustained a concussion when crashing into a wall last Friday, isn't expected to play again this season. Pinder made starts at six different positions.
Rangers: 2B Rougned Odor, who has continued to play since spraining his left ankle Sept. 10, wasn't in the starting lineup. He has started all but three games this season. But he did pinch hit and still has a chance to be the first Rangers player to appear in all 162 games since 2006, when Michael Young and Mark Teixeira both did.
A's: Right-hander Raul Alcantara (1-1), who was designated for assignment by Oakland in late April after one start, is 1-0 with a 1.42 ERA in four games (two starts) since returning to Oakland three weeks ago after Triple-A Nashville ended its season. Both of his career wins have been against Texas.
Rangers: Martin Perez (12-12, 4.83 ERA) will make his team-high 32nd start, one shy of his career high. The lefty has lost two of three starts since a career-best seven-game winning streak.
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