Swihart went 1 for 3 with a single and scored a run in his major-league debut Saturday in Boston's 4-2 loss to the New York Yankees.
He started and played behind the plate on the day the club found out Ryan Hanigan — the second starting catcher lost this year — will be out for at least two months.
"He's got age and athleticism to be able to score from first on a ball off the wall," Boston manager John Farrell said of Swihart. "It's one game to keep in mind. I think there's a reason why everyone was so attracted to him within the game and as high as we speak of him as well."
Swihart, rumored in number of deals during the offseason, was kept by the team after projected starter Christian Vasquez was lost for the year following right elbow reconstructive surgery.
Hanigan suffered a fractured pinke in Friday's loss to the Yankees. He was placed on the 15-day disabled list on Saturday.
"He had surgery today. There were some pins inserted. It's going to be a lengthy recovery time," Farrell said. "I wouldn't anticipate him back before the All-Star break."
Wade Miley (1-3) took the loss despite a solid start, holding New York to three runs in seven innings.
"He made it easy on me," Swihart said. "We had a long conversation and had a game plan going in."
Swihart learned about Yankees' top-notch reliever Dellin Betances the hard way, taking a called third strike to end the game.
"He was pretty good," he said. "That's the first time I ever faced him."
The Yankees have bounced back after a 3-6 start and lead the AL East. The Red Sox are 5-9 in their last 14 games.
Gardner doubled home a run in the third and gave the Yankees a 3-1 lead with a two-run single in the fifth. Chris Young hit his sixth homer of the year, a solo shot in the ninth.
Eovaldi (2-0) allowed two runs in 6 2-3 innings. He left after his only walk, to Swihart. Chris Martin took over and Mookie Betts hit an RBI double that just missed clearing the 37-foot Green Monster for a tying home run.
Betances then relieved and struck out all four batters he faced for his first save.
Rodriguez hit a sharp single to left field in the eighth off Alexi Ogando after grounding out three times against Miley.
Both starters pitched well after their worst outings of the season last Sunday. Eovaldi gave up four runs in 4 1-3 innings in a 6-4 win over the New York Mets. Miley allowed six earned runs in 2 1-3 innings in an 18-7 loss to Baltimore.
On Saturday, Miley gave up a run in the third when Didi Gregorius singled, took second on a wild pitch and scored on the double by Gardner, who was thrown out trying to reach third.
The Red Sox tied it in the fourth on Dustin Pedroia's fifth homer of the year.
Chase Headley led off the Yankees fifth with a double and stayed at second when Gregorio Petit's fly to center bounced in front of Betts. After a sacrifice by Gregorius and a groundout by Jacoby Ellsbury, Gardner singled.
NAPOLI CONTINUES TO STRUGGLE
Mike Napoli went 0 for 4 and his average dipped to .160. He struck out against Betances to end the eighth.
Yankees: RHP Ivan Nova allowed an infield hit and struck out one in 1 1-3 innings at extended spring training in his first game since elbow ligament replacement surgery a year ago. "More than I was expecting," Nova said. "Everything was good. I'm happy." Nova, scheduled to throw two innings Thursday at extended spring, appears on target to be back in June.
Yankees: Adam Warren (1-1, 4.35 ERA) pitches against Boston for the second time this season in Sunday night's series finale. In his first outing, he allowed one earned run in 5 1-3 innings in an 8-4 loss in Boston on April 11.
Red Sox: Joe Kelly (1-0, 4.94) faced Warren in that game and gave up one run and one hit in seven innings. He has not gotten a decision in his last three starts and allowed five runs in each of his last two.