The Belgium international, long thought to have been on the radar of Chelsea, sealed a move to Old Trafford from Everton on Monday.
Di Matteo, who was assistant coach at Stamford Bridge when Lukaku arrived from Anderlecht in 2011, believes that the striker has earned his chance to challenge for silverware after proving he can make it at the top level.
Speaking to talkSPORT, the Italian said: "Lukaku has earned his move to a club that is aiming for titles and trophies – that has always been his aim. He's an extremely hard working and ambitious young man and I think it's a really wonderful signing for Man United.
"I absolutely think they can achieve great things together. Lukaku just fits perfectly there. Jose Mourinho knows him quite well as well, and I think he will improve him further and make him a better player too.
"He works on his game every day and wants to improve. He was always asking me to do a bit more with him even after training had finished, so we did that. He has an incredible drive to succeed and have success and he's at the right place now."
Lukaku has scored 85 goals in five full seasons in the Premier League, becoming one of only five players in Premier League history to hit 50 goals before the age of 23.