Brook Lopez had 33 points, Jrue Holiday had 25 points and 13 assists and Khris Middleton finished with 26 points and 13 rebounds for the Bucks, who grabbed a 3-2 series lead and are one win away reaching the NBA finals for the first time since 1974.
Milwaukee won despite playing without two-time NBA Most Valuable Player Giannis Antetokounmpo, who is out with an injured knee. Antetokounmpo watched the game from the sidelines, standing with his teammates almost the entire game.
In his absence, four Bucks starters scored more than 20 points.
"We came out tonight and just did our best to dominate at both ends of the floor," said Lopez. "We did a great job of playing together. When everyone is scoring and everyone is doing their thing it is tough on their defence."
The best-of-seven series now switches back to Atlanta for game six on Saturday. The winner of the series will face the Phoenix Suns, who eliminated the Los Angeles Clippers on Wednesday to reach the finals for the first time in 28 years.
Role players and reserves were forced to step into the limelight Thursday as both clubs were missing their top scorers.
Antetokounmpo injured his knee in a loss in game four, and Hawks star Trae Young sat out his second straight game after he suffered a bone bruise in his right foot in game three.
Bogdan Bogdanovic scored a team-high 28 points for the Hawks in the road loss.