Both players were offered new contracts at Bristol, whom they helped win promotion to the top flight last season and to a recent upturn in form.
Eadie, 24, has made 126 Bristol first-team appearances since 2010, while Ford-Robinson, 23, has played 36 times.
Ex-Scotland Under-20 man Eadie scored against Saints earlier this term.
"Both Jamal and Mitch put in impressive performances when we played Bristol at the start of January but we have been watching them for some time," said Saints director of rugby Jim Mallinder.
"We've liked what we've seen and believe that they have the ambition and quality to have a big impact here at the Saints."
Bristol head coach Mark Tainton said he was "very disappointed" that the pair would be leaving at the end of the season.
"Both players would have played a significant role in developing what we are building at Bristol Rugby," he added.
Eadie said the decision had been "one of the most difficult" he has had to make.
"I've come through the academy system and Bristol is the club that gave me my chance in the game," he continued.
Ford-Robinson, who came through the Leicester academy, has made seven Premiership appearances so far this season.
"It's been a hugely enjoyable two years at Bristol and the support from the rugby community in the city has been fantastic," he said.
"The supporters have been magnificent towards me and I appreciate how welcome they have made me feel here."
Northampton Saints are to sign number eight Mitch Eadie and front-rower Jamal Ford-Robinson from Bristol for the start of the 2017-18 season.