The two composer's graves were opened earlier this year and it was found they were missing their teeth, both real and false.
Austrian authorities decided to act following reports that other graves might have been disturbed, coupled with the release of the disturbing video.
They have also ordered checks on the graves of other famous composers buried in the same cemetery, including Beethoven, Schubert and Schoenberg. When cemetery officials first alerted police about the raids, they decided it was not worth investigating and any crime fell outside of Austria's statute of limitations.
'The bottom line is that they just didn't seem to feel that any damage had been done,' reported one local paper. But now the Federal Criminal Police Office, the Austrian equivalent of the FBI, has been given the task of tracking down the thief.
If caught, he faces a host of charges which could lead to years in prison, including burglary and disturbing the peace of the dead.
The graveyard the composers are buried in is so famous it has its own tram, the number 71, for tourists to visit it.
And 'He has taken the number 71 tram' has even become a euphemism in Vienna for when someone has died.
Johann Strauss II died in 1899 and his beloved wife Adele erected his tombstone in his memory.
His most famous works include The Blue Danube, the Kaiser-Walzer and the Pizzicato Polka. Among his operas, Die Fledermaus The Bat and Der Zigeunerbaron The Gypsy Baron are the most acclaimed by his followers.
Brahms died in 1897. He is regarded as one of the great Romantic composers and is most famous for his Wiegenlied, better known as Brahms's Lullaby.
- Daily Mail