Anisimov, 31, appeared in 78 games, finishing with 15 goals and 22 assists, last season, his fourth with the Blackhawks. He has scored at least 20 goals four times in his career, including his first three seasons in Chicago.
Smith, also 31, has played his entire 11-season NHL career with the Senators and has been an alternate captain since the 2017-18 season. He appeared in 70 games last season, finishing with nine goals and 19 assists.
Anisimov and Smith and both signed through the 2020-21 season with their contracts carrying an average annual value of $4.55 million and $3.25 million, respectively.
"Artem is a big, competitive center with excellent hands and playmaking ability," Senators general manager Pierre Dorion said in a statement. "He plays a 200-foot game and will contribute to our depth on both special teams. We're acquiring an experienced player who can score and at the same time is responsible in his own zone. We really like how he will fit with the makeup of our team."