Behich's powerful volley from a corner kick took a deflection before flying into the top corner of the net after only 12 minutes to give the home side an early 1-0 lead.
An own goal from Karabukspor's Baris Basdas added insult to injury for Popovic's side just before the half time break.
Karabukspor fought back in the second half with Mustafa Yatabare cutting the defecit with an 81st minute finish.
It wasn't enough though and Popovic suffered his fourth straight defeat and fifth from seven games since taking over as head coach in October.
Behich's Bursaspor move into sixth spot on the Turkish Super League ladder with the win, while Karabukspor remain in the relegation zone in 17th place.
It was a big night for Australian players in Germany with Mathew Leckie, Brandon Borello and Robbie Kruse all in action.
Leckie was a key contributor in Hertha Berlin's 2-0 victory away from home against Cologne before being substituted in the 85th minute.
In Bundesliga 2. Borello had a busy night for Kaiserslauten despite his side going down 2-0 at home to Arminia Bielefeld.
The former Brisbane Roar winger played 74 minutes of the contest but collected a yellow card early in the second half.
Socceroos winger Robbie Kruse played the full 90 minutes for Bochum as they played out a 1-1 draw with Erzgebirge Aue.
In India Erik Paartalu had a huge performance scoring two goals in his side Bengaluru's 4-1 win against Dehili Dynamos.
Paartalu fired in with his head in the 24th and 45th minute as Bengaluru clinched an emphatic win to move to the top of the table.
In Belgium it was a tough day at the office for former Sydney FC shot stopper Danny Vukovic as Genk suffered a 2-0 loss to Standard Liege.
Awer Mabil played the whole match for Pacos Ferreira as they went down 2-1 to Portuguese powerhouse Sporting CP.
Socceroos defender Milos Degenek helped his side Yokohama F Marinos secure a point against Vegalta Sendai as the two sides played out a 2-2 draw, while also in the J-League Mitch Duke played 90 minutes in Shimzu S-Pulse's 3-2 loss to Albirex Niigata.
And in Norway Alex Gersbach came on as a 70th minute substitute as his side Rosenborg lost 1-0 to Oddo.