The England Under-20 international, who had been linked with a number of Premier League clubs, has agreed a four-and-a-half year deal at the Liberty Stadium.
Grimes completed his medical on Friday but will not be eligible to feature for Swansea until the league visit of West Ham on January 10.
The 19-year-old made his Exeter debut in August 2013 and went on to make 62 appearances for the Grecians, scoring five goals.
Grimes' form at the League Two club saw him become the first Exeter player to be called up by England Under-20s earlier this season along with keeper Christy Pym.
The teenager's arrival marks Swansea's second January signing after the arrival of Portugal striker Nelson Oliveira, who has joined on loan from Benfica until the end of the season.
Source : PA
Source: PA