Stramaccioni was promoted from youth team coach to guide the first team after Inter sacked Claudio Ranieri in March.
And the 36-year-old, who had no prior managerial experience in Serie A, has now put pen to paper on a three-year contract with the Nerazzurri.
"He is young and talented," Moratti told Corriere dello Sport.
"I feel it is right that he has the way and the time to be able to organise his job.
"We believe in him, we want him to build the future of Inter and this contract, in my opinion, is the right way to give him a sense of confidence from our part."
When Stramaccioni was put in charge, Inter were in eighth place after winning just one of their previous 10 games. He led Inter to a sixth-place finish.
Stramaccioni, a native of Rome, is the club's fifth coach since the departure of Jose Mourinho to Real Madrid in 2010.
After winning a string of titles under Roberto Mancini and Mourinho, the Nerazzurri have been shut out in successive seasons and what makes this year's setback even worse is that the club did not finish high enough in the table to seal a place in the 2012-13 Champions League.
Inter captain Javier Zanetti enjoyed working with Stramaccioni and believes he has a bright future.
"These months will just be the start of a long coaching career for him," Zanetti said.
"It's true that he doesn't have much experience working with first teams, but from day one he came across as a talented coach.
"He's careful about all the small details and he's very galvanising."
Inter Milan winger Wesley Sneijder is also delighted to see Stramaccioni remain in charge.
"Fortunately, we have now a coach that understands me," he said. "He gives me energy and he has put things right."