Rivera tied the major league saves record, earning the 601st of his career and matching Trevor Hoffman while preserving the New York Yankees' 7-6 win over the Toronto Blue Jays on Saturday.
"To me it was normal," Rivera said.
With his team still trying to clinch the AL East, Girardi wasn't hearing any of that.
"Right now, they're all big for us," he said.
The 41-year-old Rivera was perfect in pitching the ninth inning for his 42nd save of the season, leaving him on the brink of sole ownership of the mark.
"602 is the big one because it just puts the final stamp on it that he's the greatest closer of all time," Girardi said.
If Rivera doesn't get the record Sunday, he'll likely reach it sometime during an eight-game homestand that begins Monday.
"I would love to get it at home, but I don't think like that," Rivera said.
Elsewhere in the American League it was: Tampa Bay 4, Boston 3; Oakland 5, Detroit 3; Cleveland 10, Minnesota 4; Baltimore 6, L.A. Angels 2; Texas 7, Seattle 6; and Kansas City 10, Chicago 3.
At Toronto, Rivera struck out Colby Rasmus looking to start the ninth, then retired Brett Lawrie on a broken-bat grounder to first. While some in the crowd of 39,288 headed for the exits, most stayed to watch as Rivera got Eric Thames on a routine fly to centre.
Even before the ball had been caught, Rivera walked off the mound with his head lowered to shake hands with catcher Russell Martin. Rivera broke into a smile during the post-game congratulations line and embraced teammate Alex Rodriguez.
Centre-fielder Curtis Granderson flipped Rivera the souvenir ball.
Rivera, whose wife and two youngest sons joined him in Toronto for the weekend, told the media hordes gathered around his locker that he doesn't enjoy the spotlight of his record pursuit.
"I am uncomfortable because I don't have this much attention at all," he said, smiling. "I would like you guys to just leave it alone. That would be good."
Rivera, now in his 17th season in the big leagues, recorded his 600th save at Seattle on Tuesday. He also has 42 saves in post-season play.
"Mo is awesome," Rodriguez said. "You just run out of things to say about him. Every save he's had in his career meant something, and tonight was another example of that."
Blue Jays manager John Farrell called Rivera "the standard by which all are compared."
"If you're a fan of baseball and the game, you appreciate how good he is, how consistent he is," Farrell said.
The tying save came in Rivera's 60th appearance of the year, surpassing former Yankees teammate Mike Stanton with a record 14 seasons of 60 games.
Granderson hit his 40th homer, a go-ahead shot in the seventh to cap a 12 pitch at-bat. He fouled off seven straight pitches before homering.
Rodriguez hit his 16th home run as New York ended a two-game skid. The Yankees trailed 6-1 before beginning their comeback with four runs in the sixth.
Aaron Laffey (3-2) pitched one inning for the win. Scott Proctor, Laffey, Hector Noesi and Rafael Soriano each worked a scoreless inning before Rivera closed it out.
Bartolo Colon, who has not won in eight starts, allowed six runs and seven hits in four innings for New York.
"It wasn't great," Girardi said. "Six runs in four innings is not what we expect from Bartolo. We'll see if we can get him back on track his next start."
The Yankees closed to 6-5 in the sixth. Granderson led off with a double and scored when Rasmus bobbled Mark Teixeira's single to centre. Robinson Cano was hit by a pitch and Rodriguez followed with his first home run since Sept. 4.
"I came up in the same situation a couple of innings before and didn't come through so I was glad to get another opportunity," Rodriguez said. "I got a good pitch to hit and capitalized."
Derek Jeter opened the Yankees seventh with an infield single and Granderson homered against Carlos Villanueva (6-4).
The Blue Jays scored four times in the second against Colon. Rasmus and Jose Molina each hit RBI doubles, Lawrie drove in a run with a groundout and Mike McCoy capped it with a squeeze bunt.
The Yankees answered with a run in the fourth, an inning marked by Cano's baserunning blunder. After hitting an RBI double, Teixeira moved to third when left fielder Adam Loewen dropped Cano's fly ball to left. One out later, Nick Swisher hit a liner into right centre but Rasmus made a fine running catch for the second out. Thinking the ball would drop, Cano rounded third and passed Teixeira, who was heading back to tag up. Cano was ruled out, and the Blue Jays got credit for a double play.
"I think Robby thought the ball was going to drop for sure," Girardi said. "When Swish hit it, I thought it was going to fall, too. Still, you've got to make sure."
At Boston, Ben Zobrist hit a two-run homer off Jon Lester to lead Tampa Bay and trim the Red Sox's lead in the AL wild-card race to three games.
Jeff Niemann pitched five solid innings for Tampa Bay, which has won seven of its last eight against Boston. The Red Sox have lost 10 of their last 13 games since Sept. 3, when they led the wild-card race by nine games.
Joel Peralta pitched the ninth for his fourth save.
At Oakland, Calif., Gio Gonzalez pitched seven innings for his 14th win as the Athletics turned three double plays in the first five innings to beat AL Central champion Detroit.
At Minneapolis, Carlos Santana hit his 25th home run and Jeanmar Gomez pitched effectively into the seventh inning as the Cleveland Indians sent the Minnesota Twins to their seventh straight loss.
At Baltimore, the Los Angeles Angels' shaky playoff hopes took another blow when Ervin Santana yielded five first-inning runs in a loss to the Orioles.