Edwards is expected to sign a contract Wednesday morning and join his new teammates at practice later in the day.
Edwards' base pay is $1.65 million and the Bills already have paid him $291,176. The Jaguars would owe Edwards the balance of that deal.
The only other team that put in a claim for Edwards was the New York Jets, a source told ESPN Senior NFL Analyst Chris Mortensen, but the Jaguars were awarded the former Bills quarterback on the basis of their 1-2 record.
The Jaguars waived linebacker Alvin Bowen to make room for Edwards on the roster.
The move gives Edwards a fresh start after things went south in Buffalo and provides Jacksonville with a viable option behind Garrard.
Edwards started the first two games for Buffalo this season, completing 29 of 52 passes for 241 yards and a touchdown. He also threw two interceptions and lost both games, prompting coach Chan Gailey to turn to Ryan Fitzpatrick to spark an inept offense.
With Fitzpatrick under center, the Bills managed 374 yards -- more than Edwards mustered in his two starts. Gailey released Edwards on Monday.
Edwards could get a second chance in Jacksonville, especially if Garrard continues to perform as he has the past two weeks.
Garrard threw four interceptions and was benched in a 25-point loss at San Diego. He was even worse Sunday at home, completing 13 of 30 passes for 105 yards and an interception in another 25-point loss to Philadelphia. He also was sacked six times and botched a pitch to Maurice Jones-Drew.
Coach Jack Del Rio said Garrard seemingly lost confidence following the sixth loss in his past seven starts. He also made it clear how he felt about his quarterback's play.
"As long as he's our best option, we'll continue to go with him," Del Rio said Monday.
Those sentiments are not really new. Del Rio and team owner Wayne Weaver challenged Garrard in January. Del Rio called Garrard a "second-tier quarterback," and Weaver said he needed to be more accountable and spend more time working with the team's young receivers.
Garrard responded with what coaches and teammates called his best offseason. He even threw three TD passes in the opener. But poor decisions, bad throws and costly turnovers plagued him the past two weeks.
Jacksonville had few options, too.
Backup Luke McCown replaced Garrard in San Diego, but tore a knee ligament late in the game and was placed on injured reserve. The Jaguars signed 38-year-old Todd Bouman, but went with him only because he spent the 2008 and 2009 training camps with them and knew the offense.
Edwards' arrival could signal the end of Garrard's tenure in Jacksonville.
Garrard signed a six-year, $60 million extension in 2008. It was the richest contract in franchise history and included $18 million guaranteed. It also was a reward for 2007, when Garrard emerged as a legitimate starter, threw 18 TD passes, just three interceptions and led the team to its first postseason victory since 2000.
But Garrard has been mediocre since. He had 15 TD passes and 13 INTs in 2008, and the Jaguars won five games. He had 15 TD passes and 10 INTs last year, and they won seven games. He was sacked 42 times in each season.
Edwards has thrown for 5,739 yards, with 25 TD passes and 27 INTs.
Edwards, a third-round draft pick by Buffalo in 2007, started 32 games for the Bills. He has thrown for 5,739 yards, with 25 TDs and 27 INTs for a 76.8 passer rating.
Information from The Associated Press was used in this report.