France-born El Alagui will spend the rest of the season in Scotland, where he previously had a spell with Falkirk.
The 28-year-old's 28 goals for Falkirk helped win the Scottish Challenge Cup in 2012 and earned him a move to Griffin Park that summer before injury curtailed his first season in England.
This term has brought six goals in 18 appearances and Tannadice manager Jackie McNamara, whose side are fourth in the SPL, said: "This is the ideal time for us to bring in Farid.
"He scores goals, has strong aerial ability and will hold the ball up and allow his fellow attackers to get involved. We have moved quickly to secure him because there was interest from other clubs and I am delighted to sign him.
"He will bring something a little different from what we have at the club and will enhance our attacking threat."