The US$900,000 contract is for the 2022-23 season.
Brannstrom, 23, established new career highs in assists (14), points (14) penalty minutes (30) and games (53) with the Senators last season.
He also played in nine games for the American Hockey League's Belleville Senators.
Brannstrom had a goal and two assists in 10 penalty minutes for Belleville.
A native of Eksjo, Sweden, Brannstrom has two goals and 29 assists with 71 penalty minutes over 116 NHL games, all for Ottawa.
Brannstrom helped Sweden to a silver medal at the 2018 IIHF World Junior Championship in Buffalo, N.Y., with a goal and three assists.
This report by The Canadian Press was first published Sept. 5, 2022.
The Canadian Press