Both teams struggled from the field and were sluggish during the fourth period, but Green's foul shot lifted the Hornets into a four-point lead.
Monty Williams coached his first game with New Orleans, having come to the Hornets off the staff in Portland. The Hornets went 37-45 last year and dismissed Jeff Bower as coach.
David West scored 22 points to lead the Hornets and Marco Belinelli, who spent last season with Toronto, had 18.
Both sides shot 42 percent from the field and they combined to make 12-of-40 attempts from 3-point range.
Carlos Delfino had 19 points for Milwaukee, which fell to Atlanta in the opening round of the playoffs last year.
New Orleans had the lead throughout the fourth period, but never by more than five points.