Crawford suffered the injury and was placed on the disabled list on June 18. Prior to running first to second, and then first to third, Crawford also took early batting practice.
"He had a really good day. It's been about three days in a row now," said Red Sox manager Terry Francona. "We'll keep letting him do that and I would think in the not-too-distant future we'll be talking about him coming back. When that is, I don't know, but he's doing real well."
"We want him to be able to play and not think about running. That's so much a part of his game," added Francona.
With the All-Star break less than a week away, the Red Sox also have to be concerned with when to activate Crawford from the DL.
"If he's ready on Sunday, we would probably wait," Francona said. "If he's ready before that, (general manager) Theo (Epstein) and I need to sit down and talk to Carl, which we will do. If you're going to play one game when you're supposed to have another four days (off) you'll take the four days."
Also, pitcher Clay Buchholz will travel to North Carolina Wednesday morning to be examined by back specialist Craig Brigham. Buchholz and the Red Sox thought it best to seek a second opinion for his lower-back strain because it's not healing as quickly as he had hoped.
And injured infielder Jed Lowrie (left shoulder strain) continues to take ground balls, but has yet to take batting practice.
Lowrie has been on the disabled list since June 17.
"He still has that discomfort where he doesn't want to hit," Francona said. "But his strength is really improving, which is good."
Joe McDonald covers the Red Sox for ESPNBoston.com.