Ortiz was 0-for-4 prior to that at-bat, but as he tends to do, he delivered when the Sox needed it most.
"That's David," said Jon Lester, who allowed three runs (two earned) in 7 2/3 frames. "Even when he's kind of off, he's struggling a little bit, he always finds a way to pick us up."
"He stays at peace mentally in those key spots, and doesn't miss his pitch when he gets it," manager John Farrell said.
Lester exited with two outs in the eighth and runners on first and second, and Burke Badenhop -- who began the day with an 18-inning scoreless streak -- gave up three straight RBI singles to make it 6-4.
Then, in the ninth, Koji Uehara surrendered solo home runs to Stephen Vogt and pinch-hitter John Jaso for his first blown save of the season. He had converted 31 in a row before Sunday -- 38 including the postseason.
"I think we all get spoiled a little bit from him," Lester said. "He is human, and they put a couple good swings on him."
Luckily for Uehara and the Sox, Ortiz deposited a 1-2 curveball from Fernando Abad over the wall to start the 10th.
Uehara came back for the bottom half and redeemed himself, retiring A's closer Sean Doolittle -- a former first-base prospect who was in the game due to an injury to catcher Derek Norris -- on a ground ball to second for the final out.
"I knew beforehand that I was going to go in once we get the lead [in the 10th], so I was getting myself prepared," Uehara said through translator C.J. Matsumoto.
"He got us out of hot water, always," Ortiz added. "Got to do something for him sometime."
Before Sunday, the Sox's entire pitching staff seemed to be bailing out the offense daily -- not the other way around.
The Red Sox had plated just six total runs in the first three games of the series, and they had gone eight straight contests without scoring more than three.
But they broke out for seven runs on 13 hits on Sunday, when anything less would not have sufficed.
"We've been on the cusp many, many days, and it hasn't always worked out," Farrell said. "There's been mounting frustration, but today was a good day to bunch some hits together. That's the biggest thing."
Jonny Gomes went 3-for-5 with two RBIs, setting the tone in the first inning with a two-out, two-run single off left-hander Tommy Milone.
David Ross hit a solo shot in the second for his first homer since May 3, and Mike Napoli added another homer in the fifth.
Napoli contributed in a less characteristic way, as well, getting credited with a steal of home in the third. Jonathan Herrera was picked off first by Milone with two outs, and Napoli broke to the plate from third. The throw home was in time, but Napoli evaded the tag of Norris.
He became the first Red Sox player to hit a home run and steal home in the same game since Rico Petrocelli, on Sept. 9, 1967.
"I've got to try it," Napoli said. "I came in there [to the dugout] and said I did some ninja move to get under the tag."
"It's a big run at the moment," said Farrell. "Every run turned out to be a big one the way this one finished."
Herrera also had a strong all-around performance at short, cutting down a run at the plate and hitting an RBI triple in the eighth.
It was a team victory, and one that Boston needed in the worst way.
"Despite the last three days, our guys are still fighting, they're still putting together as tough at-bats as they can," said Farrell. "This was a hard-fought series, and it's good to salvage one out of it."