The 22-year-old completed the move from the Premier League club to the Championship outfit on Thursday.
The Gambian attacker will add to the firepower at Forest and help the Reds make a late push for the Championship playoffs.
Barrow, who was starved of playing time at Swansea, is delighted to have joined Dougie Freedman’s side, as he made it clear on Twitter earlier today.
Barrow joined Premier League side Swansea from Ostersunds FK in the summer of 2014 and could make only seven appearances in the English top flight.
The 22-year-old did not score a single goal and created five chances in those games, while having a pass accuracy of 78%.
Forest are currently ninth in the Championship table with 53 points from 36 matches, nine points behind sixth-placed Brentford.