The rise of streaming has fractured the television audience, and it has become a rarity for a show to draw more than 20 million viewers. The exceptions are live broadcasts of major sporting events and awards shows.

The interview segment with Ms. Clifford made for a bigger draw than the telecasts of this year’s Grammys (19.8 million) and the Golden Globes (19 million). The “60 Minutes” episode also had more viewers than the usual editions of NBC’s “Sunday Night Football,” which averaged 18 million viewers, and came within striking distance of ABC’s telecast last month of the Academy Awards, which had 26.5 million viewers.

The highly anticipated segment on Ms. Clifford kicked off viewing parties.

The broadcast also outperformed recent “60 Minutes” breakout episodes. The El Chapo-focused interview with Sean Penn, in January 2016, drew 20.6 million viewers, and Lesley Stahl’s sit-down with Mr. Trump and his family, soon after he won the presidential election in November 2016, drew 20 million.