Cassano pounced in the 11th minute, controlling a superb Andrea Pirlo pass before rounding goalkeeper Rene Torgard and slipping the ball into an empty net for his seventh international goal.
The Faroes fought back valiantly and were unlucky not to equalise, Suni Olsen hitting the post midway through the first half and Christian Lamhauge Holst thumping the bar with 20 minutes to go.
"We tried to create more but were just too sluggish," said Italy coach Cesare Prandelli.
"At international level anything can happen. The pitch played slow and we were missing the final pass but there are no excuses when you have so much possession."
Midfielder Joan Edmundsson shot and then headed wide before Olsen found space 20 metres out and struck a fierce drive against keeper Gianluigi Buffon's post to bring the 5,000-plus crowd to their feet.
Italy, who thumped the Faroes 5-0 a year ago, struggled to create an opening, with Daniele De Rossi's long-range shot their only serious attempt before the break.
"We've not qualified yet," said Prandelli. "We have to stay focused. We've just seen how difficult this game can be."
