The fan was identified as Shannon Stone, a 39-year-old firefighter from Brownwood, Texas.
The tragedy occurred in the second inning of the Rangers' 6-0 victory over the Athletics.
"As an organization, our team members and our staff, we are very heavy hearted about this," Rangers president Nolan Ryan said in addressing the media after the game. "Our thoughts and our prayers go out to the family. Our thoughts are with them."
The fan fell headfirst in the second inning while leaning over the front-row railing trying to catch a ball that had been tossed into the stands by left fielder Josh Hamilton. Athletics outfielder Conor Jackson had hit a foul ball that caromed out to Hamilton in left.
As most Major League players do, Hamilton flipped the potential souvenir into the stands. The fan reached over the railing, caught the ball and then fell headfirst into the opening behind the scoreboard.
"All of us at Major League Baseball are shocked and saddened over the tragic death of Mr. Stone last evening," Major League Baseball said in a statement. "Our thoughts and prayers are with his son and his entire family. Major League Baseball has the utmost sensitivity to the safety of all the fans that come to our ballparks. Our players are encouraged to be fan-friendly and we will carefully review this incident with our clubs to continue to ensure a safe environment for our fans."
Ronnie Hargis told the Dallas Morning News that he was sitting next to the man who fell.
"He went straight down," Hargis said. "I tried to grab him, but I couldn't. I tried to slow him down a little bit."
Another fan, David Dodson, told the Morning News that the man lost his balance as the ball hit his hand.
"It looked awful because you knew there was no way he was going to land on his feet," Dodson said.
The fan was immediately attended to by emergency personnel and taken to a Fort Worth hospital. According to the Arlington Police Department, the man went into full arrest on the way to John Peter Smith Hospital and was pronounced dead at the hospital.
Stone fell and landed not far from the Athletics' bullpen.
"When they had him on a stretcher and they were carrying him out, he was saying stuff like 'Please check on my son, he's left up there by himself.'" Athletics reliever Brad Ziegler said. "The people that carried him out reassured him that, 'We'll get your son, we'll make sure he's OK.' We just kind of assumed that he's talking, he's conscious, he'll be OK. To find out he's not is just ... tough."
The Rangers were aware of the incident, but did not know the extent of the man's injuries.
It was only after the game that Hamilton and his teammates found out that the man had passed away. Ryan spoke to the players and the rest of the team in a postgame meeting in the clubhouse.
"We spoke to them, and they understand what happened," Ryan said. "Josh is very distraught over this, as is the entire team."
The Rangers did not open the clubhouse to the media after the game.
"It's always difficult when someone loses their life," Rangers manager Ron Washington said.
"It filtered into our dugout, and talk about having an effect," Athletics manager Bob Melvin said. "It got through our dugout. Certainly, baseball's not very important in light of something like that."
The Rangers announced that all of the flags at the ballpark have been lowered to half-mast in the memory of Stone. The Rangers will also observe a moment of silence prior to Friday's game.
T.R. Sullivan is a reporter for MLB.com. Read his blog, Postcards from Elysian Fields, and follow him on Twitter @Sullivan_Ranger. This story was not subject to the approval of Major League Baseball or its clubs.