Terms were not disclosed, but NFL Network reported the deal is worth $15 million.
The move comes after Drew Brees restructured his contract to create more than $10 million in salary-cap space for the Saints.
Brown becomes the latest Patriots player to depart the Super Bowl champions in free agency, joining defensive end Trey Flowers (Detroit Lions), left tackle Trent Brown (Oakland Raiders) and wide receiver Cordarrelle Patterson (Chicago Bears).
Saints starting nose tackle Tyeler Davison is an unrestricted free agent and the team has depth concerns at their other defensive tackle spot, with starter Sheldon Rankins recovering from a torn Achilles and backup David Onyemata facing potential discipline for a marijuana possession charge.
The Patriots selected Malcom Brown in the first round of the 2015 draft, with the final pick of the round (No. 32). They declined his fifth-year option for the 2019 season, which would have kept him under contract for a base salary of about $8 million.
In his four seasons in New England, Brown appeared in 60 regular-season games (51 starts). Playing mostly on early downs, the 6-foot-2, 320-pound University of Texas alum was credited with 186 tackles and totaled 8.5 sacks.
In 11 playoff games, he totaled 33 tackles and one sack.
Brown, who turned 25 in February, often was asked to play a two-gap style for the Patriots in which the primary goal was to hold up blockers to allow those behind him to make plays. That was a change for him coming from Texas, where he started 26 of 39 games over three years and finished with 165 tackles and 8.5 sacks. He had 29 tackles for a loss for the Longhorns.
ESPN's Mike Triplett and Mike Reiss contributed to this report.