The selection headed to Atlanta is a conditional seventh-round draft choice, league sources told ESPN.
Saubert, who enters his third NFL season, was part of a crowded depth chart in Atlanta headlined by Austin Hooper and Luke Stocker. In New England, he joins a mix of players who are vying to help fill the void created by Rob Gronkowski's retirement.
The trade comes four days after one of the Patriots' top tight ends, Matt LaCosse, hobbled off the field in the preseason opener with an undisclosed leg injury.
The 6-foot-5, 253-pound Saubert entered the NFL as a fifth-round draft choice of the Falcons in 2017. He played in all 16 regular-season games last year, totaling five catches for 48 yards. Saubert also contributed on special teams and was viewed as a reliable utility-type option.
In college at Drake, Saubert ranked second among tight ends in his final season by averaging 70.5 receiving yards per game (56 receptions for 776 receiving yards).
In addition to Saubert, the Patriots' tight end depth chart includes LaCosse, veterans Benjamin Watson and Lance Kendricks, second-year player Ryan Izzo and former practice-squad player Stephen Anderson. Watson is suspended for the first four games of the season for violating the NFL's substance abuse policy.