The 24-year-old reportedly attracted interest from numerous clubs, including Premier League pair Burnley and Watford, but the recently-demoted Championship outfit have won the race.
While the club have opted against disclosing the fee paid, reports emerged recently claiming that Boro were willing to part with £15m.
"I can't wait to get started and train with the boys. I'm glad it's completed and I can focus on what's ahead now," Assombalonga told the club's official website.
"I can't wait to meet the manager and get ready for next season. There's a good feeling around the place. From the minute I walked in it felt right."
The striker heads to the Riverside Stadium having scored 30 goals in 47 starts for Forest in the second tier since 2014.