Redd had 14 points and Bogut added eight in the final 7:09 as Milwaukee improved to 11-1 when its 7-footer scores at least 17 points.
Chicago had a chance to tie as time expired. John Salmons inbounded to Luol Deng, who passed it back to Salmons. Salmons went up for a shot, but sent it back to Deng, who missed an open 27-footer as time expired.
Derrick Rose scored 25 points and Joakim Noah had 10 points and 18 rebounds, but the Bulls dropped their third straight after a four-game winning streak.
Milwaukee trailed 76-69 when Brad Miller made two free throws and Rose hit a layup, but Chicago went almost 7 minutes without a field goal.
That allowed Redd and Bogut to take over. Redd scored 11 straight points for the Bucks and Bogut added the next six to change a seven-point deficit into an 88-80 lead with 3:02 left.
Milwaukee, which squandered a 17-point, first-half lead, nearly lost its late lead, too.
Deng hit a jumper, Rose added a layup and Miller hit a 3 to cut it to 93-91 with 1:01 left.
After Hakim Warrick slammed an offensive rebound to make it 95-91 with 10.1 seconds left, Rose added a layup and Redd hit the second of two free throws to set up the Bulls' final chance.
The game almost got ugly with 10:12 to go when Kirk Hinrich and Bogut began jawing at each other and pushing and shoving ensued.
Officials assessed technicals on Hinrich, Tyrus Thomas, Bogut and Warrick, but it could've been much worse. No punches were thrown and Bulls assistants made sure no one left the bench while officials took several minutes reviewing the tape of the incident before assessing the fouls.
The Bucks had made it a focal point to win these last three games after a 4-15 stretch.
