Dortmund, who snapped up Germany left back Nico Schulz on Tuesday, said 26-year-old Hazard had signed a deal to 2024.
"He is an experienced Bundesliga player and Belgium international who can help us with his speed and finishing," said Dortmund sports director Michael Zorc.
"He has continuously displayed his quality in the past years," he added of Hazard, brother of Chelsea's Eden, who joined Gladbach in 2014.
He scored 13 goals this season and set up another 12 in 35 matches in all competitions but fifth-placed Gladbach narrowly missed out on a Champions League spot.
Dortmund have also agreed to sign Bayer Leverkusen's talented and versatile 23-year-old attacking midfielder Brandt on a five-year deal, pending some final details.
"I am a person who sometimes decides things on as hunch," Brandt said in the Dortmund statement. "For me this feeling is important and I have a very good feeling about Borussia Dortmund. I want to further develop here."
Dortmund finished two points behind champions Bayern Munich in last week's Bundesliga finale and will compete in the Champions League group stage next season.
(Reporting by Karolos Grohmann; Editing by Amlan Chakraborty and Ken Ferris)
