Dawson took to Twitter to announce his free-agency move.
"I am humbled & thrilled to have the opportunity to continue my career with my new team, the SF 49ers!" Dawson tweeted.
San Francisco signed Dawson to a one-year deal worth $2.25 million, league sources told ESPN NFL Insider Adam Schefter.
Dawson, 38, has made 14 of 15 tries from 50-plus yards over the past two seasons. He has made 13 of 14 attempts from 40 to 49 yards over that span. Dawson will replace veteran David Akers, who came under fire last season for a subpar year.
Akers, possibly diminished after undergoing hernia surgery, made 9 of 19 attempts from 40 yards or longer last season and was released ahead of free agency.
"I haven't had any direct conversation with David, but he's a guy I hold in the highest regard, both on and off the field," Dawson said during a conference call Tuesday night. "There's worse people to follow than him. The career he's had is one we all wish we could have."
San Francisco also added depth to their new-look receiving group, signing wideout Marlon Moore to a one-year contract.
The NFC champion Niners made the announcement Tuesday. Last week, San Francisco acquired Anquan Boldin in a trade from the Baltimore Ravens, who beat the 49ers 34-31 in the Super Bowl on Feb. 3 in New Orleans.
The 25-year-old Moore, a Northern California native from Sacramento, played the past three seasons with the Miami Dolphins after joining them as an undrafted free agent in April 2010. He has 12 career catches for 244 yards and two touchdowns in 29 games.
ESPN.com's Mike Sando and The Associated Press contributed to this report.