Boro confirmed the fee at £15m on Monday night, finally eclipsing the £12.8m the club paid for Brazilian flop Afonso Alves almost 10 years ago.
Assombalonga is the fifth new face of the summer overall, and the second new striker after Middlesbrough already spent £9m on Danish forward Martin Braithwaite from Toulouse.
"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," the 24-year-old told MFC.co.uk.
"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."
Assombalonga arrives at the Riverside Stadium with a respectable scoring record at Championship level and has the potential to become one of the division's top stars. He netted 14 times in a Forest side that was almost relegated last season, while he scored 30 in 47 starts for the club since 2014.
Crucially, Assombalonga appears to have fully overcome the serious knee injury that wrecked his 2015/16 season and should now go from strength to strength in a potent Boro side.