Anderson has agreed to a three-year, $26 million contract, including $14.5 million guaranteed, to stay with the Browns, reports NFL Network's Adam Schefter.
The team had been negotiating a three-year, $20 million deal for Anderson. The 24-year-old agreed to a deal just hours after he became a restricted free agent.
Anderson threw 29 touchdown passes and 19 interceptions last season and went 10-5 after taking over as the starter in Week 2. He made the Pro Bowl as an alternate.
Savage needs to upgrade Cleveland's defensive front with only Robaire Smith and Shaun Smith currently under contract.
Anderson hit the free agent market at 12:01 a.m. after his agents were unable to complete a deal with the Browns. Savage believed other teams were interested in Anderson.
If he had signed with another team, the Browns would have had a week to match that offer. Savage indicated the team likely would have let him go and received a first- and third-round picks in April's draft as compensation.
The Browns don't have a first-round pick after trading it away last April to select Quinn, the former Notre Dame star who threw only eight passes in his rookie season for Cleveland.
Savage's preference was to keep both quarterbacks for at least another season, but with Anderson back, the Browns still have the option of trading one of the two QBs to get draft picks and improve a team that just missed the playoffs last season.
Anderson failed to beat out Charlie Frye in training camp last season, but took over in Week 2 after Frye was traded to Seattle.
The Associated Press contributed to this report