Milwaukee, which has won three of its last four, made 56.5 percent of its shots from beyond the arc.
Cousins, who returned from a 10-game absence due to viral meningitis, had a chance to win it in the closing seconds, but his jumper was off the mark as the buzzer sounded. He finished with game highs of 27 points and 11 rebounds in just under 33 minutes in the Kings' fifth straight loss.
Sacramento went 2-8 in Cousins' absence and fell to 0-2 under interim coach Tyrone Corbin, who relieved Michael Malone of the head coaching duties this week.
"(Cousins) did a great job out there in trying to will us to a win," Corbin said.
Ben McLemore went for 22 points on an efficient 8-for-10 shooting and Rudy Gay added 20 points in the setback.
While Sacramento got its center back, the Bucks were without theirs as Larry Sanders served a one-game suspension due to a Flagrant-2 foul in Wednesday's loss to Portland.
With the game tied at 81-81 entering the fourth, Cousins knocked down a pair of free throws to open the quarter before Dudley bookended an 11-1 run with a pair of baskets. Dudley's 3-pointer with 8:40 remaining gave the Bucks a 92-84 advantage.
Cousins converted a three-point play with just over four minutes to go that cut the deficit to seven, then Darren Collison followed a Knight layup with a 3 to make it a 105-101 deficit. Gay continued to keep the momentum rolling for the Kings with a floater that preceded his tough layup that knotted the score at 105-105 with 1:01 remaining.
Jerryd Bayless then drove the lane and left off a pass to Zaza Pachulia, who made the open layup to put the Bucks back in front. McLemore hit a pair of free throws at the other end to again tie the contest.
On Milwaukee's ensuing possession, Dudley missed a rushed 3, but Pachulia pulled down the rebound, was fouled and split the two free throws with 6.8 seconds on the clock.
The Kings ran the final play through their best player. Cousins gained separation from Pachulia with a step-back jumper, however, his try was short and the Bucks held on.
"We wanted to get the ball in his hands," Corbin said of the final play. "He had a good look."
The Kings got an immediate lift from their big man in the opening minutes. Cousins netted eight points in the first 5:02 of the contest as the hosts jumped out to an 18-6 lead.
Mayo proceeded to knock down a pair of 3-pointers, then his jumper with over a minute remaining in the quarter capped the scoring and brought the Bucks within 29-25 entering the second.
"With O.J. coming out, he gave us a spark offensively," said Bucks coach Jason Kidd. "He kept us in there early."
Milwaukee grabbed its first lead since the first minute of the game when Bayless hit a 3 for a 39-38 edge with under eight to play in the half. Knight's trey with 5.2 seconds left helped the visitors to a 57-54 lead at the break.
Milwaukee wraps up its four-game road trip against the Clippers on Saturday ... Sacramento went 38-for-67 (56.7 percent) from the floor ... Bayless had 17 points off the bench in the win ... The Kings committed 21 turnovers ... There were 14 ties and 10 lead changes ... Bucks rookie forward Johnny O'Bryant started for the first time in his career and finished with six points and one rebound in 17 minutes in place of Sanders.