Reserves Rodney Stuckey and Drummond each finished with 16 points, and Tayshaun Prince and Villaneuva added 14 for the Pistons.
"Look, we had opportunities to win the game," Pistons coach Lawrence Frank said. "It just didn't fall for us."
Williams, starting for the second straight game, finished with 18. Teague had 17 and Anthony Morrow 16 for Atlanta, which outscored the Pistons 36-9 on fast-break points.
The Hawks were up by 22 early in the fourth quarter when DeShawn Stevenson's 3 from the left corner made it 84-62.
But as Bynum kept driving the lane and beating the Hawks to the rim, Smith went cold for the Hawks, missing his last seven shots in the fourth and both overtime periods. Smith was 13 for 18 through the first three quarters.
Both Bynum and Smith missed potential game winners at the end of the first overtime. Smith defended Bynum on a layup, but Villanueva played a role in successfully defending Smith's left-handed hook at the buzzer.
"Even though they were making some difficult plays, I think we were scrambling out on defense," Smith said. "We switched the pick-and-roll a little bit and we were trying to make it a little difficult for them to be able to shoot over a taller defender. That's what we were able to do — limit them to one shot and get the rebound."
For the Pistons, who were trying to earn a rare three-game winning streak, the loss dropped them to 3-14 on the road, 9-22 overall.
There was little consolation for Bynum.
"I got hot, and other guys picked it up, and piece by piece things started to work," Bynum said. "We got back in the game, but I missed a key layup at the end. That was unbelievable. The missed shots haunt me now, and they were still haunting me while we were out there. We put forth a great effort, but it just wasn't enough."