Rivera recorded his 25th save on Sunday, reaching the plateau for the 15th consecutive season to extend his Major League record streak.
Rivera struggled for 16 seconds while trying to find the right words to describe the milestone.
"I'm a guy that ... I play for the team," the 41-year-old closer said. "When you play for a team like the New York Yankees, you're going to have opportunities. And all I care [about] is trying to do my job and help the team as much I can, and all those other things will come out. Yeah, 25, that's good, but I just wanted to win ballgames."
The Yankees barely did that on Sunday after All-Star setup man David Robertson uncharacteristically could not get out of the eighth inning.
Entering with two runners in scoring position and a 6-4 lead, Rivera got Cliff Pennington to ground out to second to end the threat before running into trouble of his own in the ninth.
In that frame, Rivera surrendered four straight one-out singles -- allowing one run to cut the Yankees' lead to 7-5 -- before David DeJesus lined a cutter right to Mark Teixeira at first base for a game-ending double play.
"I think one guy hit the ball hard -- line drive to [left fielder Brett Gardner]," Rivera said of the ninth. "But really, just ground balls that find a way, broken bats, find the holes, and that's the game. You really can't get upset at that, because all of a sudden, line drive, double play. That's part of the game. I don't worry about those things.
"I worry about the guy that is at home plate, try to get him out and if we don't get that one out, we'll get the next one. That's what I think."
Rivera was asked to record a four-out save just once this season before Sunday, but he blew a 3-2 lead in the ninth inning of an eventual extra-inning win on April 24 at Baltimore.
But both Rivera and manager Joe Girardi did not think the extra work was something that affected the 17-year veteran.
"I'm here to do my job, regardless if it's four outs or three outs," Rivera said. "Don't matter what it is. I have to be ready for that, and I am."
Girardi said after the game he does not think Rivera's record of 15 straight seasons of at least 25 saves will be broken.
"It's just how consistent he's been, and his longevity," Girardi said. "To think of anyone being a closer for 15 years is somewhat mind-boggling. You don't see it happen very often. And to be real effective for 15 years is another story."
Matt Fortuna is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.