However, they will not be happy with the manner of the win as they did little to show they had learned the lessons from last Friday's 2-2 draw with the Bulgarians.
Italy coach Cesare Prandelli was looking for an improved performance on their opening qualifier and things looked hopeful when Gli Azzurri took the lead after just five minutes.
Claudio Marchisio's pass from the right landed at the feet of Destro, who smashed a shot into the bottom right corner from 18 yards.
Another goal almost followed five minutes later from an Alessandro Diamanti free-kick, but Malta goalkeeper Andrew Hogg did brilliantly to tip the ball over the crossbar.
Pablo Osvaldo had the home side's next chance soon after with a header from Andrea Pirlo's corner but the Roma striker steered the ball past the left post.
Another chance fell Osvaldo's way after 25 minutes as Italy continued to drive forward in numbers but defender Luke Dimech cleared the ball for the Maltese.
Pietro Ghedin's men had barely had a look-in so far but reminded the Italians of their presence on the half-hour with a long-range effort from Michael Mifsud which flew over the bar. Gli Azzurri responded with a cross-shot from Diamanti which was easily saved by Hogg.
Italy goalkeeper Gianluigi Buffon had been surplus to requirements for much of the half, but had to get his fingertips to a right-footed strike from Roderick Briffa after 33 minutes as Malta threatened a surprise equaliser.
The momentum quickly swung back Italy's way, however, although Leonardi Bonucci and Pirlo were both wasteful with chances before the interval.
Malta keep hosts honest
Italy continued to rule the roost at the start of the second half and defender Andrei Agius had to clear a threatening strike from Lorenzo Insigne. Osvaldo also saw his volley held by Hogg in the Malta goal as another chance went begging.
And nerves began to jangle around the Alberto Braglia stadium after 60 minutes when Briffa drilled a shot narrowly over the bar for the unfancied visitors.
The midfielder unleashed another strike six minutes later which had to be booted out of the area by Antonio Nocerino, before Italy launched an attack down the other end which resulted in Marchisio firing over the bar.
The home side began to settle down once more and Bonucci put another effort high of the mark before Destro saw his effort saved by Hogg as Malta continued to frustrate their opponents.
Giampaolo Pazzini was unable to turn Gli Azzurri's fortunes either, smashing the ball past the left post, but they eventually made the breakthrough in stoppage time when Pirlo's corner was neatly headed home by Peluso.