The 22-year-old came through the ranks at Leicester City and scored a hat-trick on his senior debut in a 4-1 win against Bury in the EFL Cup in 2015.
Dodoo made a single Premier League appearance and scored again in his next EFL Cup game, before a loan move to Bury where he played five times, scoring once.
The player then went on to sign for Gers in 2016, going on to make 25 appearances in all competitions, but has not appeared for the club this season.
Charlton manager Karl Robinson told the official club site: "We have been looking for a striker that can complement what we already have.
"We have a big number nine in Josh Magennis and Joe gives us something different. I think fans will like him, he is quick, can stretch the game and get in behind defenders."