Gillespie's terrific play was reminiscent of Mays' historic catch on Vic Wertz in the 1954 World Series.
"We've all seen that play a million times," Gillespie said of Mays' catch. "I think of those ballparks, it seems like it was 500 feet out there. This is probably the closest thing to it. I've kind of made those catches in practice before."
Statcast™ projected Byrd's drive at 408 feet from home plate, with an exit velocity of 102 mph.
"It's probably one of the best catches I've made in my career, I'll say," Gillespie said. "I know in this park, there is a lot of real estate out there in center. When there is a ball hit over your head, there is a good chance it might not be going over the fence."
Gillespie noted his experience on his high school football team.
"I think my wide receiver days in high school might have helped there," Gillespie said of adjusting on the over-the-shoulder catch.
Gillespie, making his third start in center since being called up from Triple-A New Orleans, prevented Byrd from recording at least a double or maybe a triple off lefty Adam Conley, who made his first MLB start.
"I played with Cole in the beginning of the year and saw him make some outstanding defensive plays," Conley said. "He's a good player out there. That guy hit the ball pretty good. Luckily in Marlins Park there is room to make that play out there and room to run around."
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.