Saubert, 25, appeared in 16 games for the Falcons last season and made five catches for 58 yards. The fifth round pick in the 2017 NFL Draft did not appear in any games for the Falcons in 2015 or 2016. He did not make a catch in 14 games during his 2017 campaign.
The trade comes days after Patriots tight end Matt LaCosse sustained a leg injury during the team's preseason win against the Detroit Lions. New England signed LaCosse in March, just before the retirement of star pass-catcher Rob Gronkowski.
The Patriots also signed veteran Ben Watson, who is suspended for the first four games of the season, and veteran Lance Kendricks.
Watson is listed as the starting tight end on the Patriots depth chart, with LaCosse, Stephen Anderson, Kendricks, Ryan Izzo and Andrew Beck listed as backups.
The Patriots battle the Tennessee Titans in another preseason game at 7 p.m. CDT Saturday at Nissan Stadium in Nashville.