With Matt Holliday on first and one out in the sixth, Jhonny Peralta smashed one to deep left.
Cespedes raced back and snared it, holding on to the ball even as he ran into the wall.
Thanks to the catch, the game stayed in a 1-1 tie. The Red Sox went on to win, 2-1, with Cespedes opening the winning rally with a single and scoring the go-ahead run on a sacrifice fly by Xander Bogaerts.
"I did actually get a pretty good read on that," said Cespedes. "I usually like to take a peek back at the fence to see where it is when I see the ball. But I wasn't able to do that this time, and that's why the fence kind of snuck up on me."
The Red Sox acquired Cespedes last week from the Athletics in exchange for ace lefty Jon Lester and outfielder Jonny Gomes.
"I've really taken pride on my defense this year and I've worked on it a lot, especially throwing to the bases, and I think it's been evident so far this year," said Cespedes.
He is enjoying his time with Boston so far.
"I feel pretty good," he said. "Although we're not in a position to really win at this point or go anywhere in the postseason yet, it's a pretty united team, and I feel pretty comfortable here."
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, and follow him on Twitter @IanMBrowne. This story was not subject to the approval of Major League Baseball or its clubs.