Red Sox right-hander Rick Porcello hit Escobar with a fastball just above the right elbow in the bottom of the first inning. Escobar, who leads the majors with 32 doubles this season, remained in the game before leaving in the fourth. He was replaced at third base by Taylor Motter.
The Twins announced that Escobar has a bruised elbow and is considered day-to-day.
Asked whether he thought Porcello intentionally hit Escobar, Minnesota manager Paul Molitor paused before saying, "It didn't look particularly good to me."
Porcello moved into a tie for the American League lead with his 10th hit batsman.
Said Escobar, through an interpreter: "I respect Porcello a lot. He's one of the best pitchers in the league. If he did it on purpose, I don't really know what the purpose is behind it. I've never done anything against him. I was trying to protect myself, and we'll just move on."
Gibson also hit Sandy Leon with a pitch in the sixth.
Escobar, who is hitting .299 with 12 home runs and a team-leading 48 RBIs, is just 15 doubles shy of tying Justin Morneau's club record of 47, set in 2008.
The Associated Press contributed to this report.