Carter-Williams connected on 10 of 15 shots from the floor, handed out nine assists and grabbed eight rebounds for the Bucks, who narrowed the gap in their best-of-seven Eastern Conference first-round series to 3-2.
The young Milwaukee team had stayed alive with a game-four victory over the Bulls thanks to a buzzer-beating layup by Jerryd Bayless.
And their second straight win over the Bulls was more convincing as Milwaukee led for all of the second half.
“(The team) worked their tails off and put themselves in position to make this a series,” Bucks coach Jason Kidd said. “No one had us in this position. We just go out and play basketball.”
The Bulls slashed their turnovers to 13 after 28 in game four, but shot a disappointing 34.4 percent from the floor — connecting on a dismal seven of 30 attempts in the fourth quarter.
Former NBA Most Valuable Player Derrick Rose scored just 13 points on five-of-20 shooting.
The Bucks will now try to even the series when they host game six on Thursday, but they still face an uphill climb. No team has rallied from an 0-3 deficit to win an NBA playoff series.
The Bucks, in the playoffs a season after posting the worst record in the league, weren’t the only team battling the odds on Monday.
In Portland, the Trail Blazers rallied for a 99-92 victory over the Memphis Grizzlies to extend their Western Conference first-round series to a fifth game.
Damian Lillard led the Blazers with 32 points, C.J. McCollum added 18 points and LaMarcus Aldridge added 18 points and 12 rebounds for the Blazers, who denied the Grizzlies a four-game sweep.
Marc Gasol led the Grizzlies with 21 points, as the Spaniard contributed with seven rebounds and six assists.
Courtney Lee chipped in 19 points and Zach Randolph contributed 12 points and nine rebounds for a Memphis team playing without star point guard Mike Conley.
Conley had facial surgery on Monday to repair a broken bone suffered in the third quarter of the Grizzlies’ game three win.
Elsewhere in the East, Brooklyn Nets point guard Deron Williams dazzled with 35 points to lead the eighth seeds to a 120-115 overtime victory over top-seeded Atlanta that tied their series at two games apiece.
Williams, who totalled only 18 points in the first three games of the series, tied career playoff highs for points and field goals made at 13.
“It feels great to get the win, that’s the most important thing,” Williams said. “This was a team win all the way. We fought and we scrapped. There was never any let up.”
Brook Lopez added 26 points and 10 rebounds, Joe Johnson scored 17 and three other Nets players finished in double digits.
They will try to take the lead in the series when it returns to the Hawks’ home court for game five on Wednesday.