The Fiorentina forward glanced in a Mauro Camoranesi cross in the 78th minute, tapped in from close range three minutes later and then got the last touch on a deflected Fabio Quagliarella shot in injury time.
The Cypriots had looked on course for a memorable win after going 2-0 up with Ioannis Okkas's 12th-minute strike and Chrysostomos Michail's volley just after the break.
The hat-trick capped a memorable few days for Gilardino, who also scored the last-gasp goal that earned Italy a 2-2 draw in Ireland on Saturday to book their tickets to next year's finals in South Africa as Group Eight winners.
With qualification secured, Italy coach Marcello Lippi changed his entire starting XI.
His side lacked intensity as a result before the late rally and took some flak from the home crowd, which in turn prompted a furious reaction from Lippi after the match.
"This team are world champions. On Saturday we qualified for the World Cup with a match to spare and naturally we celebrated and we weren't as focused and determined as if it had been a decisive game," he told Rai television.
"I changed the whole team against and it is normal for us to have some problems against a side that beat Bulgaria 4-1 on Saturday.
"But when there was the first difficulty, part of the crowd got on these lads backs and called out the names of players who aren't here. Are we mad? I think these lads deserve more respect."
Fabio Cannavaro returned from suspension to captain Italy after being cleared of a doping charge earlier this week.
