(CNN) The batteries dying on a TV remote control? We've just about all been there.

And Roger Federer -- one of the greatest tennis players of all time and the fifth highest-paid athlete in the world, according to Forbes -- said it happened to him when he was trying to watch Naomi Osaka and Coco Gauff's now famous interview at the US Open on Saturday.

In a poignant moment that won't soon be forgotten, defending US Open women's champion Osaka comforted the 15-year-old sensation Gauff after she beat her 6-3, 6-0 in the third round.

Osaka then convinced Gauff -- who gained worldwide attention and masses of fans by making it to the fourth round at Wimbledon in July -- to take part in the on-court interview with her, something usually only reserved for the winner.

Federer took note and wanted to listen in.

"I was having dinner. ... I was just watching it from afar in the second set," the Swiss told reporters after crushing David Goffin in the fourth round Sunday. "Then I realized they were chatting to one another. I figured as they were both going up for the interview, let me listen in, what have they got to say?

"I couldn't get the volume to work because the remote had no more battery. I'm trying to figure it out. Watching an interview with no words is kind of hard for me.

"I got it working at the end. I caught the back end of it. I saw highlights and all that."

And what did Federer make of the interview?

'Tennis won on the night'

"I just thought it was a nice moment," he said. "It showed both personalities from both players. In a way also tennis won on the night, not just Osaka. Also Coco, the way she handled herself anyway. It's incredible at 15. I can't even think being so well behaved at 15.

"Of course, everything comes crashing down on you. You're exhausted. It's finally over. There's even relief. You're sad about the result. I thought it was a big moment.

"There was huge momentum since Wimbledon with Coco here in the States. It was a great ending in some ways, that interview, rather than maybe just a 6-0 set in the second set, walking out, just getting a great ovation. I think it was a nice way to show that the players care for one another on this tour."

It's not quite over yet for Gauff, though.

She and fellow American 17-year-old Caty McNally, who took a set off Serena Williams in the second round in singles, upset ninth seeds Nicole Melichar and Kveta Peschke 6-3, 7-6(9) in the doubles second round Sunday in a packed Louis Armstrong Stadium.

"The crowd was unreal today," Gauff said. "I didn't think it would be that many people at the match, to be honest. I mean, I knew we were on Armstrong, but I couldn't see an empty seat and they were sold out and so supportive. I think that definitely helped us, especially when the pressure moments came."

Gauff and McNally are the reigning junior doubles champions at the US Open and won a top-level title in Washington last month. They've yet to drop a set as doubles partners, and the teenagers have dubbed themselves as "McCoco."

Their next opponent will be No. 8 seeds Ashleigh Barty and Victoria Azarenka.