The 22-year-old attacking midfielder, who has played for the Denmark Under-21 team, made four Champions League appearances for Nordsjaelland this term.
"We did not sign him for his name but because of his high potential. His skills convinced us," St Etienne president Roland Romeyer told the club's website.
The deal includes an option for a permanent move.
Andreas was born in Barcelona where former Denmark and Barcelona star Michael won four consecutive La Liga titles from 1991-94 and the first European Cup in the club's history in 1992 under coach Johan Cruyff.
Laudrup junior had a spell at Real Madrid's academy, the club his father joined in 1994 before clinching a fifth Spanish title a year later.
St Etienne, who are 10th in Ligue 1, host 12th-placed Toulouse on Friday.