Aloisi had previously worked under Muscat as an assistant at Japan's Yokohama F. Marinos.
Former Socceroo Muscat was appointed as Shanghai Port's new head coach earlier in December.
"Leaving Brisbane Roar has been a very tough decision for me," said Aloisi, 50, who joined the Roar in April on a two-year deal.
"I wholeheartedly believe in the club's progress and direction.
"However, the opportunity presented to me overseas is something that aligns closely with my personal journey and goals as a coach.
"It's a path I feel I must take at this stage in my career."
The fifth-placed Roar's last match with Aloisi at the helm was a 3-0 home loss to Central Coast Mariners on Thursday.
Roar CEO and chairman Kaz Patafta said the club respected Aloisi's decision.
"We did not want him to leave at this time, but we also did not want to stand in the way of an opportunity that is rarely presented to Australian coaches," Patafta said in a statement.
"Our club is on a remarkable upward trajectory, and we will continue to build on this momentum.
"We are in the process of finalising key contract extensions, which will be announced in the coming weeks.
"These extensions are a clear indicator of our players' belief in the club's direction and future."
Aloisi's assistant Luciano Trani has been promoted to interim Roar head coach until the end of the 2023-24 season.
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