Smith, who remained in Atlanta after being discussed in talks leading up to Thursday's trade deadline, came through with his defense one day later.
Al Horford led Atlanta's balanced scoring with 24 points and the Hawks overcame a slow start to beat Sacramento 122-108 on Friday night.
Smith also had six rebounds and four assists, but coach Larry Drew talked the most about the decision made by the 6-foot-9 power forward to defend the 6-foot-11, 270-pound Cousins, the Kings' high-scoring center.
"Josh signaled he did not want any help down there," Drew said.
"Josh did a phenomenal job on him and that's what Josh has been since he's been here. Josh is the type of guy, he will take the challenge against a guy like that."
Teague said Smith "can guard anybody, one through five, and that's what makes him so special."
Smith said Thursday he didn't sleep well Wednesday night as the trade deadline neared. He said he "slept too good, actually" on Thursday night and had the energy to hold Cousins, who had 26 points and 13 rebounds, to only four points in the final quarter.
"I just wanted to show him different looks," Smith said. "Whenever you show a guy different looks, it kind of wears him down. I just wanted to be able to use my quickness on him."
Smith will be an unrestricted free agent after the season.
"I'm glad I'm here," Smith said. "We've got to think about moving forward and keep pushing for better playoff position."
Isaiah Thomas scored 30 points with nine assists to lead the Kings, who have lost four straight and eight of 10.
Tyreke Evans chipped in 15 points but no other Sacramento player scored in double figures.
Sacramento led by 13 in the opening period. The Kings were up 63-61 at halftime but couldn't keep pace in the second half as their road struggles continued. The Kings have lost seven straight on the road and are 5-24 away from Sacramento.
"This is an offensive-minded team," Cousins said. "We're struggling in all aspects of the game. I do believe our offense carries our defense.
"We went away from what worked. We have a tendency to do that every game. I mean, if you haven't learned yet, you ain't trying to learn."
After 14 lead changes, the Hawks took the lead for good on Teague's 3-pointer with 2:20 remaining in the third quarter.
Atlanta pushed its lead to double figures for the first time at 105-94 on Stevenson's fifth 3-pointer midway through the final period.
Kings coach Keith Smart said the Hawks' 33-19 advantage in fast-break points was evidence the Sacramento players weren't hustling on defense.
"That's effort and energy," Smart said. "The effort defensively wasn't there."
For one half, the Kings' hot shooting was enough.
Led by Thomas, who had 12 points in the first quarter, the Kings made 10 of their first 14 shots from the field and took their biggest lead at 31-18.
"You're feeling pretty good, you feel like you're going to win the game with your offense," Smart said.
The Hawks took control by outscoring the Kings 32-25 in the third quarter.
Sacramento's John Salmons said the Kings, who "focus too much on offense," should stop using youth as an excuse.
"The whole league's young at this point," Salmons said. "The young thing's not really an excuse anymore. It's time to grow up, be a team."
The Kings had only 10 players as they awaited the arrival of the three they acquired from Houston on Thursday: forward Patrick Patterson, center Cole Aldrich and point guard Toney Douglas. Sacramento gave up forwards Thomas Robinson, Francisco Garcia and Tyler Honeycutt in the deal.
The Kings expect the three newcomers to join the team Saturday in New Orleans for Sacramento's next game on Sunday.
Atlanta's Dahntay Jones, acquired from Dallas for Anthony Morrow on Thursday, was in uniform and played the final 2 minutes. Center Jeremy Tyler, purchased from Golden State at the trade deadline, was with the team but was not active.
Sacramento's Jason Thompson had nine points and 10 rebounds.
Smith's buzzer-beating 3 to end the first half was disallowed following a long review. Officials said the clock, which was set at 0.8 seconds, started late.
NOTES: The Kings have lost 10 straight against Atlanta since their last win in the series on Feb. 20, 2008. ... The Hawks wore red-and-black uniforms from the 1990s as they recognized their all-1990s team selected by fans. Kevin Willis, Dikembe Mutombo, Steve Smith and Dominique Wilkins were honored at halftime. Mookie Blaylock, the fifth member of the all-decade team, was not present. ... The Kings' last road win was Jan. 28 at Washington. ... Stevenson's five 3-pointers matched his season high.