Zverev won the final five points to defeat Anderson 6-7, 7-5, (10-7). His victory helped Team Europe beat Team World 13-8.
"Obviously it was an amazing feeling," Zverev said. "So special to clinch it for us guys and Europe. It's great how it ended."
Anderson, who lost in the Wimbledon finals this year, said Sunday night's setback affected more than just him.
"You're playing for something different. You feel like you didn't come through for the team," he said.
Had Anderson won, Nick Kyrgios of the World team would have played Wimbledon champion Novak Djokovic in a deciding singles match. But it was not necessary because the European team had reached 13 points.
Roger Federer earlier put Team Europe ahead by saving three match points for a 6-7, 7-6, (10-7) victory over John Isner.
Isner staggered Federer with a body serve that caused the 20-time Grand Slam winner to stumble early in the match. But Federer regrouped and fought off two match points in the second-set tiebreaker before capturing the deciding 10-point tiebreaker used in the Laver Cup.
In the opening doubles match, Jack Sock and Isner saved two match points in the deciding tiebreaker to beat Federer and Zverev 4-6, 7-6, (11-9).
The three-day, five-session tournament at the United Center drew 93,584 fans.
Team Europe won the first Laver Cup last year in Prague. The tournament moves to Geneva, Switzerland, next year.