Joshua - the super-heavyweight champion at the 2012 Olympics - landed a series of heavy blows against the Italian who looked completely out of his depth from the outset.
Leo was previously unbeaten in eight fights, but Joshua knocked him down with a stunning flurry of blows just two minutes and 47 seconds into the contest.
The 32-year-old Leo had fought only eight times over the course of over four years before the bout and never before outside his native Italy. He was also giving away notable height and weight advantages to the 17-stone Joshua.
The Brit was understandably delighted after his dominant performance and ranked it alongside his Olympic victory at London 2012.
"It was as important as the Olympics and everyone's proud we got the victory," Joshua told Sky Sports.
"I'm really happy with my performance. You only get one pro debut and we did it in spectacular fashion.
"It's just the start - a beginning - but we can build on that."
The 23-year-old from London will next fight in Sheffield on October 26.
On the same bill, fellow Olympic champion Luke Campbell produced a first-round knockout of Darlington's Neil Hepper in the lightweight division.
In what was only Campbell's second paid fight, the man from Hull disposed of his opponent in one minute and 59 seconds.