Bridgewater found out about the trade and boarded the team bus to say goodbye to his former teammates, who were heading to Philadelphia for Thursday's preseason game, a source said.
"In his time with us, Teddy was the consummate professional, a great teammate and a tremendous leader," general manager Mike Maccagnan said in a statement on the team's website. "He worked diligently with our medical staff and made tremendous strides along the way. We wish him nothing but success in the future."
The Jets went into training camp looking to trade Bridgewater, 25, whom they signed to a one-year, $6 million contract. They've already paid him $1 million in bonus money.
Bridgewater will cost the Saints at least $5 million for one year. He could earn up to $9 million more in incentives based on playing time, yards and touchdowns.
"We are excited to add Teddy Bridgewater to our team," Saints coach Sean Payton said in a statement on the team's website. "We look forward to working with him and watching him continue to develop as a player."
Bridgewater played well in three preseason games, completing 74 percent of his passes (28-for-38) for 316 yards, two touchdowns and one interception. His passer rating is 104.7.
The trade occurred one day shy of the two-year anniversary of Bridgewater's horrific knee injury with the Minnesota Vikings. Bridgewater played in 29 games for the Vikings in 2014 and 2015 but missed all of 2016 after dislocating his left knee and tearing multiple ligaments on Aug. 30 of that year. He has recovered remarkably and hasn't been limited in any fashion. He played in one game last season and signed with the Jets in March.
The Saints appear to have decided to upgrade at backup quarterback behind Drew Brees this year. But Bridgewater also is an intriguing candidate to eventually succeed Brees, if the two sides ultimately decide to extend Bridgewater's contract past this season.
Brees, who will turn 40 in January, just signed a two-year deal with the Saints in March.
The Saints have never made this kind of investment in the backup quarterback position during the 13-year Brees-Payton era, although they did come close to drafting Patrick Mahomes in 2017. The Saints are paying a significant price to land Bridgewater, considering they have less than $10 million in salary-cap space and already traded away their 2019 first-round draft choice to move up to pick defensive end Marcus Davenport this year.
Savage was solid but unspectacular throughout the preseason.
The performance of Hill, meanwhile, was more of a roller-coaster ride. He led the Saints in rushing in each of the first three preseason games, but he also committed four turnovers in the first half of the second exhibition game.
Hill will almost certainly remain on the Saints' 53-man roster because they're intrigued by his long-term potential and because he also serves as one of their top special-teams coverage specialists.
Information from ESPN's Mike Triplett was used in this report.