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.
A lethargic Italy held on and with 19 points from seven games, eight ahead of Serbia and Slovenia, the group leaders will guarantee a place in the finals by beating Slovenia on Tuesday.
"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."
Prandelli sent out a strong side against the part-timers, again pairing Cassano and Giuseppe Rossi up front, and they almost took the lead within a minute through Riccardo Montolivo.
It was not long before the in-form Azzurri, who defeated world and European champions Spain in their last outing, made the breakthrough through Cassano.
The Faroes reacted well and created three chances in as many minutes midway through the first period.
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.
Giampaolo Pazzini, on for the disappointing Rossi, added energy to the Azzurri attack in the second half but it was the home side who again came closest to scoring with 20 minutes to go when Holst fired against the bar with Buffon beaten.
Prandelli threw on Mario Balotelli to stir up his side but despite dominating possession late on, Italy rarely threatened a second goal.
The Italians have now gone a year without conceding a goal in competitive matches.
"We've not qualified yet," said Prandelli. "We have to stay focused. We've just seen how difficult this game can be."
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