The 26-year-old defensive midfielder has signed a five-year contract and could make his debut against Schalke at the weekend.
Many Arsenal fans hoped Gunners manager Arsène Wenger would entice the Brazilian to the Emirates Stadium, such is the squad's need for a defensive midfielder.
Gustavo has opted to remain in Germany
However Gustavo has opted to stay in Germany, where he will earn over €100,000 a week with his new club.
Gustavo made 136 Bundesliga appearances both for Bayern and former club Hoffenheim, scoring 11 goals.
Following the arrivals of Mario Götze and Thiago Alcântara this summer his chances of appearing regularly were reduced significantly and Wolfsburg made their move a few weeks ago.
Gustavo played in both legs of Bayern's last 16 win vs Arsenal this year
Gustavo initially said he would prefer to stay with Bayern and fight for his place, but having featured only fleetingly under new coach Pep Guardiola in pre-season, he has had a change of heart.