The natural disaster hit several days before the anniversary of a major quake that killed nearly 300 people in central Italy near Amatrice.
Seven people were also missing after a building collapsed and 25 were slightly injured.
The two toddlers were said to have had bottles of water passed to them by the rescue workers. The children were reportedly aged at 6, 4 and 7 months old.
Healing facility authorities said every one of the three were doing great, with the more established two young men being dealt with for parchedness and the most seasoned for a break to his correct foot. Media reports said one woman had been killed when she was hit by falling rubble from a church.
The quake "was really strong, and really scary", local councillor Donatella Migliaccio told news agency DPA.
It is the height of the tourist season on Ischia, and Italian television showed visitors fleeing bars, hotels and restaurants to take refuge in parks when the undersea quake struck. Dozens of people are said to have ended their holidays early and Italian media said some tourists were sleeping on benches waiting for boats to arrive.
Most of the damage was in the high part of the volcanic island.
After a sleepless night filled with the noises of helicopters and sirens, the couple - who were travelling with Rosanna's brother - planned to cut their stay on the island a day short and leave after the quake.
The INGV had initially classified the quake as a 3.6 with a 10km-deep epicentre, raising questions about how such a relatively weak seismic event could wreak so much havoc.
It caused several buildings to collapse before residents and tourists ran out on to the busy streets.
He suggested that the destructive effect may have been amplified by local geological factors or by shoddy constructions "built without any earthquake-proofing".
Together with the nearby island of Capri, Ischia is a favorite island getaway for the European jet set and German Chancellor Angela Merkel has been known to visit at Easter.
Two women were killed in the 4.0-magnitude quake while 39 were injured and 2,600 left homeless.
Italy sits on the boundary between the Eurasian and African tectonic plates, making it vulnerable to seismic activity whenever the large plates collide.