The Red Sox lost the opener, 6-5, but took the nightcap with an offensive surge, winning 18-9.
"He took a swing, the very first swing he took down the cage and his neck just spasmed up," manager Terry Francona said of Crawford. "He tried to loosen it up and went about his normal day. It just wasn't happening. It was getting worse."
As for Crawford's availability for Game 2, Francona said earlier, "I don't know, we'll see."
Monday's first game was the second consecutive game Crawford did not start. On Sunday, the Sox started Conor Jackson instead of Crawford in left because of Crawford's struggles against Tampa Bay starter David Price. Crawford did enter the game later as a pinch hitter and doubled.
Kevin Youkilis also did not start either game Monday as he continues to battle hip bursitis and a sports hernia.
Darnell McDonald took Crawford's place in left field for Game 1 Monday and batted ninth.
Joe McDonald covers the Red Sox for ESPNBoston.com.