Swihart led off the 10th inning with an inside-the-parker and the Red Sox stopped the New York Mets' seven-game winning streak, holding on for a 6-4 victory Friday night.
The Red Sox, visiting the Mets for the first time since 2001, scored three times in the 10th. Swihart's drive appeared to clear the center field wall and bounced back into play, but the ball was ruled in play.
"I thought it had gotten over because of the way it bounced back, but I just kept my head down running," Swihart said. "I kind of watched the center fielder jogging after it, but I didn't hear anything so I kept running."
There was no indication from the umpires that it was a home run, and the ball caromed back past center fielder Juan Lagares.
"I don't think he planned on an inside-the-park home run," interim manager Torey Lovullo said. "But we run hard until our base coaches stop our guys and he did exactly what he was supposed to do."
The fleet-footed Swihart kept going as shortstop Ruben Tejada ran far into the outfield to retrieve the ball.
Approaching third base, Swihart got the green light from coach Brian Butterfield, and easily beat second baseman Wilmer Flores' relay throw.
Lovullo indicated that the team planned on having the play reviewed if Swihart had not crossed the plate.
Swihart, who went 2 for 4, also stole a base during Boston's three-run seventh inning.
"I pride myself on being an athletic catcher," Swihart said. "I have the ability to steal bags and when the opportunity arises to get into scoring position, I'm going to do it."
The NL East-leading Mets rallied in the bottom half when reliever Junichi Tazawa walked four straight batters with two outs to force home a run. Craig Breslow replaced Tazawa and retired Yoenis Cespedes on a fly ball with the bases loaded for his first save.
"Our bullpen is spent," Mets manager Terry Collins said. "When we went into extra innings, it was tough to decide where we were going to go."
Mets ace Matt Harvey pitched six shutout innings, giving up two hits and striking out eight. He left with a 2-0 lead.
Ortiz exited with left heel tightness in the eighth. Lovullo said they would reevaluate the slugger before Saturday's game.
Mets captain David Wright went 2 for 5 in his first game at Citi Field since April. He returned from a long stint on the disabled list earlier in the week during a sweep in Philadelphia.
The crowd gave Wright a standing ovation when he came to bat for the first time.
Boston rookie Henry Owens gave up one earned run in five innings.
Red Sox: OF Hanley Ramirez (shoulder) was a last minute scratch. Lovullo indicated that Ramirez was dealing with right shoulder fatigue and was day-to-day. Earlier in the season, Ramirez sprained his left shoulder. ... Lovullo said that Tazawa had some "lower back stiffness" after fielding d'Arnaud's double-play ball in the 10th.
Mets: INF Daniel Murphy didn't start. Collins indicated that several other regulars would be given a day off over the weekend. Collins also indicated there was a "good shot" that Wright would sit on Saturday.
Red Sox: Joe Kelly (7-6, 5.18 ERA) is coming off his longest outing of the year and has thrown three consecutive quality starts for the first time all season. Kelly is 1-0 against the Mets in three career appearances.
Mets: Jacob deGrom (12-6, 2.29 ERA) looks to rebound from the worst start of his career. DeGrom allowed a career-high seven runs in just 2 2/3 innings against Philadelphia.
Tampa Bay Rays place center fielder Kevin Kiermaier on COVID-19 related IL
The Tampa Bay Rays placed center fielder Kevin Kiermaier on the COVID-19 related IL due to general illness symptoms and purchased the contract of right-hander Joey Krehbiel from Triple-A Durham before Saturday's game against Detroit.
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Gray scheduled to start for Cincinnati against Los Angeles
Los Angeles Dodgers (94-54, second in the NL West) vs. Cincinnati Reds (77-71, third in the NL Central)
Howard expected to start as Rangers host the White Sox
Chicago White Sox (84-63, first in the AL Central) vs. Texas Rangers (54-93, fifth in the AL West)
Burnes expected to start for Milwaukee against Chicago
Chicago Cubs (66-82, fourth in the NL Central) vs. Milwaukee Brewers (90-57, first in the NL Central)