Despite his recent arrest, the 23-year-old started during Sunday's game against Indianapolis. He had five tackles in a 27-7 loss to the Colts.
Smith was arrested Friday on suspicion of DUI and marijuana possession.
He faces a suspension from the league, but the team has stepped in to take control of the situation.
"I don't know that there's a right decision in this situation," 49ers CEO Jed York said. "It was obviously something that was very disappointing to the team and I think in talking with Aldon and knowing what he wants to do, there's not a one-day fix. This is a long-term fix. We support Aldon if Aldon is willing to fight for himself and try to make this right and I will stand here and I will support him and I'll take any shots that anybody wants to direct at the organization, you can direct them to me. And I will support Aldon as long as he's willing to work at this and fight to get better."
Smith is a key pass rusher and sacks leader for the Niners, and coach Jim Harbaugh had said leading up to Sunday that the NFL would decide how to punish the third-year pro, who turns 24 Wednesday.
Smith did not field questions, but apologized after the game, "First off I wanted to apologize to the team, the organization, my family and everybody I let down. I also wanted to let it be known that this is a problem and it's something that I will get fixed, and that I'll do everything in my power to make sure that this never happens again. I also wanted to let everybody know that once again I'm sorry and, like I said, this won't happen again."
He'll be arraigned in court in November.
(The Associated Press contributed to this story)