"Bressan has signed a 3-1/2-year contract with Alania. He will become a full member of our team after completing a medical," the club said on their website.
The Brazil-born player has impressed many pundits both at home and in Europe with his excellent skills and scoring ability this season.
Bressan, 24, helped BATE clinch their seventh consecutive league title two weeks ago, becoming the team's leading scorer with 11 goals despite playing mostly in midfield.
He also scored two of BATE's eight goals in the Champions League, helping a modest club with limited resources beat Bayern Munich and Lille as they secured third place in Group F and a spot in the Europa League knockout round with one game to spare.