Braden pitched the 19th perfect game in Major League history on Sunday, a 4-0 blanking of the Rays, needing just 109 pitches to retire all 27 Tampa Bay batters.
Rodriguez opened the weekend at Fenway Park by saying that he would not be drawn further into a war of words with Braden, who had criticized A-Rod again in a recent interview. He then prepared to close the three-game series against the Red Sox by praising him.
"I've learned in my career that it's always better to be recognized for some of the great things you do on the field," Rodriguez said in a brief statement during batting practice. "Good for him. He threw a perfect game, and even better he beat the Rays."
The perfecto bid was a topic of discussion during Yankees manager Joe Girardi's pregame availability with reporters, some of whom were checking for updates on their wireless phones.
"I hope he gets it," Girardi said as Braden prepared to head into the ninth inning. "A perfect game is an amazing feat. I've had the chance to watch one and catch one. I don't have any hard feelings toward Dallas Braden.
"As I've said, there's a competitive edge to people, and we don't play them until July [5-7], so I'm not really worried about it. You know, I hope he gets it. God bless him."
The fact that the perfect game would come against the Rays, a team the Yankees trailed by a half game in the American League East entering play on Sunday, was an added bonus.
"That always helps too, being within our division," Girardi said, with a chuckle.
Behind closed doors, the Yankees had largely found humor in Braden's claims that A-Rod violated an unwritten code of baseball on April 22, when he passed over the mound on his way back to first base following a foul ball.
Braden shouted at Rodriguez during and after that inning, then continued the discussion later that evening through the media. The next day, Rodriguez said that he found the whole hubbub "pretty funny" and that "we've been laughing about it since."
Some still were. As he prepared for his start Sunday at Fenway, Yankees starter A.J. Burnett spotted highlights of Braden hugging his grandmother, Peggy Lindsey, who was present for the perfect game in Oakland.
"Grandma, I love you, but don't cross my mound!" Burnett cracked.
Brian Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.