Hagelin agreed to a two-year, $4.5 contract. He will earn $2.1 million next season and $2.4 million in 2014-15. He carries a salary-cap hit of $2.25 million.
The 24-year-old Falk, acquired by the Rangers from Minnesota on June 30, will earn $975,000 on his one-year contract.
Hagelin, who is also 24, had 10 goals, 14 assists and a plus-10 rating in playing in all 48 games last season. He ranked sixth on the team in points and goals and third in shots. New York went 14-2-1 when he had a point.
In the playoffs, Hagelin had career-highs with three goals, six points, a plus-six rating, and 35 shots in 12 games.
Falk had three assists and 40 penalty minutes in 36 games with Minnesota last season.