Azarenka, who lost 7-5 in the third set to Williams in last year's final.
"We gave it everything we got."
In the men's doubles final, Leander Paes and Radek Stepanek were convincing winners over Alexander Peya and Bruno Soares.
The India-Czech pairing of Paes and Stepanek dominated the title match to record a 6-1 6-3 victory over their Austria-Brazil rivals.
Paes, 40, won his third U.S. Open doubles title and eighth major doubles title of his career. It was Stepanek's second major doubles win after claiming the 2012 Australian Open with Paes. Stepanek, 34, said the win enabled them to qualify for an important goal.
"There is one thing we would like to achieve, and that's winning the world championships at the end of the year, because that's the trophy which is missing in Leander's showcase," the Czech said.
Croatia made a clean sweep of the junior singles titles when Borna Coric and Ana Konjuh posted victories at Flushing Meadows. The fourth-seeded Coric beat Australia's Thanasi Kokkinakis 3-6 6-3 6-1 in the boys' final before second-seeded Konjuh beat U.S. wildcard Tornado Black 3-6 6-4 7-6(6).
Coric, 16, said the U.S. Open would be his last junior tournament.
"I've won the slam" he said about his Sunday victory. "That was the goal at the beginning of the year." Coric has been named to Croatia's Davis Cup team for the tie next week against Britain at Umag.