But Sunday night, the high-priced left-hander has a chance to send the Yankees packing on the wrong end of a four-game sweep when he faces Masahiro Tanaka in the series finale.
The Red Sox have won the first three games of the series with their rivals by a combined 23-9, increasing their lead atop the American League East to a season-high 8 1/2 games.
Now, they turn to Price, who is 15-13 lifetime against the Yankees but 2-6 with a 9.21 ERA against them since joining the Red Sox.
This season, he has faced the Yankees twice, allowing 12 runs, including six homers, in 4 1/3 innings.
"It's time for me to kind of go back to that drawing board and kind of reinvent myself against these guys," Price said after getting hammered July 1 at Yankee Stadium.
Price (11-6, 3.97 ERA) hasn't lost since that start, going 2-0 with two no-decisions in July. In his last three starts, he has allowed four earned runs and struck out 18 in 21 innings.
But those starts weren't against the Yankees.
In his three years in a Red Sox uniform, he went 1-3 with a 7.89 ERA in 2016, 1-1 with a 4.15 ERA in 2017 and then 0-2 with a 24.52 ERA this season.
Tanaka (9-2, 3.84) has not lost since April 17, going 7-0 with six no-decisions since. In his last two starts, he has worked 15 scoreless innings, yielding six hits while striking out 17.
He is 8-4 with a 4.18 ERA lifetime against the Red Sox, 4-2 with a 4.09 ERA at Fenway. He is 1-0 this year but with a whopping 8.71 ERA -- and he has allowed four home runs in 10 1/3 innings.
The Red Sox (78-34) are 44 games above .500 for the first time since 1946 -- and they came within one out of back-to-back shutouts. In fact, New York was scoreless with four hits in 17 innings against Rick Porcello and Nathan Eovaldi before scoring a run with two outs in the ninth inning against Craig Kimbrel in a non-save situation Saturday. It was Kimbrel's first action since July 29.
Greg Bird hit a lazy fly ball with the bases loaded to end the game, dropping the Yankees to 68-41 and spoiling the debut of Yankee rookie right-hander Chance Adams.
"I'm a broken record. Tomorrow is always important," New York manager Aaron Boone said. "We're wearing it right now from the Red Sox. We're in a tough spot. Tomorrow is important. This is the big leagues. We're a really good team. We've got to right the ship. Tomorrow's important. But if we had won five in a row, tomorrow's really important.
"Look, there's no question that they've established themselves right now as the best team in this league. That's indicative of their record and how consistent they've been. Obviously, they do a lot of things really well.
"That said, if you walk through our room right now, to a man, we know we can absolutely play with them. We know that when we're at our best we can beat them. We acknowledge who they are right now. There's no denying the season that they're having."
Eovaldi has thrown 15 scoreless innings in his two Boston starts, the first Red Sox pitcher to throw at least that many in his debut with the team since Bill Rohr in 1967.
"He is who we thought he was," Boston manager Alex Cora said.
J.D. Martinez is 7-for-14 with three home runs and new Boston hero Steve Pearce 5-for-15 with two homers against Tanaka, but Jackie Bradley Jr. is 5-for-30, Eduardo Nunez 3-for-15, Mookie Betts 8-for-36 and Brock Holt 6-for-25.
The Red Sox placed newly acquired second baseman Ian Kinsler on the 10-day disabled list with hamstring tightness, summoning Tony Renda from Pawtucket, where he was hitting .333.
Betts, who came up as a second baseman but is a Gold Glove-winning right fielder, was back at second for the first time since 2014 for most of Friday's game and was thrilled to be there. He was back in right field Saturday.
"It was like a dream come true," Betts said after Friday's game. "Haven't been there since 2014. That's why I take my ground balls, just in case."
"He works at it," Cora said. "He takes ground balls. He's an athlete. This wasn't a regular thing. Ian comes out and we want to stay away from Xander (Bogaerts) and Ron (bench coach Roenicke) wanted to put him at second and I was OK with it."
The Yankees made two minor deals Saturday, acquiring right-hander George Kontos from the Cleveland Indians and infielder Giovanny Urshela from the Toronto Blue Jays. Both players were sent to Triple-A Scranton/Wilkes-Barre.