The 18-year-old's contract was due to run out at the end of the current A-League season.
The Australia Under-23 international has been a highly sought after acquisition, with a number of clubs around Europe known to be interested in the left-back, but it now appears he will ply his trade in Norway's Tippeligaen.
Rosenborg are Norway's most successful team with a staggering 23 league titles, 13 of them between 1991 and 2004.
They have played in the UEFA Champions League group stage 11 times and famously drew with Chelsea in September 2007, a game which signalled the end of Jose Mourinho's first spell at the club.
Gersbach, who played his last game for Sydney in Saturday's 3-1 defeat at home to Brisbane Roar, will fly out on Monday to undergo a medical and tie up any remaining loose ends on his contract.
The fee is undisclosed but is believed to be in the region of $500,000.
Gersbach's departure is a sizeable blow for Sydney FC, who don't appear to be in a rush to sign a replacement.
However, Rhyan Grant has been a square peg in a round hole when asked to fill in at left-back on occasion this season and Riley Woodcock hasn't been sighted since his move from Perth Glory.
Ali Abbas could be another option for Sydney coach Graham Arnold when he works his way back to full fitness.
Arnold said the opportunity was too good for Gersbach to resist.
“Alex is very ambitious young man and I would like to thank him for his service to Sydney FC,” said Arnold.
“It is always pleasing to be able to bring through a young player, develop him and see him get a chance in Europe.
“We would have loved to have kept him for another year but it’s a great opportunity for him and I wish him the very best for the future.”
Gersbach himself described his excitement at fulfilling his European dream and said he will always be a Sydney FC fan.
"The chances I've been given at Sydney FC have undoubtedly helped me improve as a player and presented me with this wonderful opportunity which lies ahead. I am very grateful for that," he said.
"I've had a great two years with an exceptional group of players and coaching and medical staff, and it has been an honour to represent this club and its fantastic supporters.
"I am hugely excited about the next stage in my career and I wish the club every success in the future. I will always be a Sydney FC supporter at heart."