Aubameyang, 28, has been subject of negotiations between the Gunners and Dortmund for several weeks, with Arsenal having refused to meet the Bundesliga side's €70m asking price.
The striker, who scored 31 league goals last season, was left out of two consecutive games for Dortmund after missing a team meeting, with head coach Peter Stoger accusing the frontman of not being focused.
However, Aubameyang started in Dortmund's draw with Freiburg on Saturday amid claims from the Bundesliga side that a transfer would be sanctioned if Arsenal reached "certain parameters".
Arsenal eventually reached an agreement to make Aubameyang a club-record signing after Dortmund wrapped up the loan of Michy Batshuayi on the final day of the January transfer window to replace him .
Dortmund had been keen on bringing Olivier Giroud to the Westfalenstadion as a short-term replacement for Aubameyang but the France international opted to stay in England for family reasons. Following Batshuayi and Aubameyang's moves, Giroud could make the switch to Chelsea before the window closes.
Aubameyang joins up with former Dortmund team-mate Henrikh Mkhitaryan at Emirates Stadium after the Armenian joined in a swap deal which saw Alexis Sanchez head to Manchester United. The duo combined for 62 goals in all competitions two seasons ago.
This season, Aubameyang has scored 21 goals in 24 matches in all competitions. He has found the net 141 times since joining BVB from Saint-Etienne for €13m in July 2013, and he also has 23 goals in 56 caps for Gabon.
"Apart from the unpleasant events of recent weeks, we like to remember that the story of Pierre-Emerick Aubameyang at BVB for more than four years was a single success story," sporting director Michael Zorc told Dortmund's website.
"He has done great things for Borussia Dortmund in this time, scored many important goals and is part of the team that brought the DFB-Pokal to Dortmund in 2017. We wish Pierre-Emerick all the best for his future at Arsenal."
He joins an Arsenal side that sit sixth in the Premier League table, five points off the pace in the race for the top four.
Arsene Wenger's side were eliminated in their FA Cup third round meeting with Nottingham Forest earlier this month but have advanced to the Carabao Cup final against Manchester City and will face Swedish side Ostersunds in the Europa League last 32.