Roma forward Destro struck his first goal for the Azzurri in the fifth minute while Atalanta defender Federico Peluso's last-gasp header was deflected into the net by Malta's Cohen in injury time.
Cesare Prandelli had reacted to his side's "insufficient" 2-2 draw against Bulgaria on Friday by making five changes and replacing a tricky 3-5-2 tactical formation with a 4-3-1-2.
Malta, who sit 133 places behind Italy in FIFA's world rankings in 139th, frustrated the beaten Euro 2012 finalists, notably in a second half when the hosts were guilty of squandering a number of chances.
But after claiming September "has never been a good month for the national team", Prandelli was delighted to pick up the three points.
"We have to look at the positive aspects, in the second half we were a far more disciplined unit," said Prandelli.
Malta's Italian coach Pietro Ghedin said: "We played solidly against a really good team. Everyone expects Italy to perform wonders, but we closed them down."
The match gave Prandelli the chance to tinker and his decisions were soon vindicated.
Destro, who replaced Juventus frontman Sebastian Giovinco to team up with Roma colleague Pablo Osvaldo, ran on to Claudio Marchisio's perfectly-timed ball to beat 'keeper Andrew Hogg from close range.
"When you play with certain teammates, you can score in every match because they know how to set you up," said Destro.
"I'm really happy to have scored. I'm young, and I'm happy to play in different positions if the team want me to."
Bologna playmaker Alessandro Diamanti, another new inclusion, then saw his superbly-taken free-kick from the right tipped over by an alert Hogg.
Osvaldo, who scored both Italy's goals against Bulgaria, was then unlucky to see his header from a corner go wide under pressure, while only a fine tackle from Michael Mifsud saved Malta when Osvaldo looked poised to score his third goal for Italy after rounding Alex Muscat.
Malta's only real chance of the half came when Daniel Bogdanovic set up Roderick Briffa, whose well-struck drive was tipped over by Gianluigi Buffon.
Prandelli replaced Diamanti with Lorenzo Insigne at the interval and the little Napoli forward was a constant threat for the Maltese defence, whom he teased with mazy runs down the left wing.
Despite his good service to Italy's forwards, however, they failed to convert as Malta dug deep in a bid to hold on.
On the hour mark Osvaldo, too, was left frustrated when, after Insigne had made space on the left, he took the ball out of play instead of laying off for the unmarked Roma man.
Despite their domination, Italy toiled at times and gave Malta space which the visitors were ultimately unable to exploit.
At the other end, Marchisio spurned a great chance when his long-range effort flew over Hogg's crossbar.
Osvaldo then came off for AC Milan striker Giampaolo Pazzini, who twice spurned chances to put the match beyond reach.
In the final minutes, Prandelli sent on Giovinco for Destro, but it was defender Peluso's header, from an Andrea Pirlo corner, that finally gave Italy the chance to breathe easy when it bounced off Cohen and into the net.