It was the Americans' second straight group stage victory, and they will play the Czech Republic for a spot in Saturday's final.
Berdych had expected to face Lleyton Hewitt, but the former No. 1 withdrew because of a stomach problem. Berdych struggled against Ball and had to save eight of the nine break points he faced before closing out the match with an ace.
"You are just preparing all day to play Lleyton Hewitt, you get some tactics or whatever, strategy, and then just change it in the last moment," Berdych said. "So, it was a quite tough one. He was playing well."
In the Blue Group, France beat Serbia 2-1 and will play Argentina for a spot in the final. The Argentines overcame Germany by the same score, despite trailing 1-0 after Tuesday's opening singles match.
Eduardo Schwank evened the series by beating Germany's Andreas Beck 7-6 (3), 2-6, 6-3, before Juan Monaco and Horacio Zeballos teamed to beat Christopher Kas and Philipp Kohlschreiber 6-2, 6-3 in the doubles.
The final round-robin matches will be played over two days, with the winner of each group advancing to the final.