Elise never saw her coming.

She was still in awe, staggering in a dream-like state along deck five of the Disney Wonder — an entire deck designed with lower ceilings to make it feel cozier to children on Disney’s first cruise ship to sail out of Miami. Then, someone snatched the black-and-white fedora off my hipster-cool 9-year-old’s head.

A Disney Wonder crew member, in her late 20s and wearing the identifiable yellow polo shirt, cackled as she ran off with Elise’s hat. Elise’s cool was busted as she gave chase. Soon, my two other daughters, Amelia, 7, and Catalina, 5, joined in hot pursuit, all of them quickly embroiled in an impromptu game of keep-away.

They were having such a good time — even the crew member — and the ship hadn’t yet left the port.

For the Disney aficionados who have not yet sailed with the Mouse, this should come as refreshing and enticing news: To take a Disney cruise is to get the full Disney experience on the high seas.

The Wonder, the Mouse’s second of four cruise ships, built in 1999, has been repositioned from Los Angeles to Miami, where it began sailing in late December and now does regular four- and five-day tours to the Caribbean. Disney has committed to two more seasons of Caribbean cruises out of Miami.

A Florida-born kid, I consider myself a Disney connoisseur since it was my family’s annual vacation throughout my childhood. My own children grew up with a possibly unhealthy devotion to the Disney princesses in specific, and to the parks in general.

There is something to that “Disney magic.” In my experience at the parks and on the ship, it’s called service. If anything sets Disney apart, it’s the ability to provide a service a guest doesn’t yet know he wants.

You don’t have to be a parent to identify the gene that produces Disney magic. But you’ll know it when you see it.

For me, it was on my first dinner on the ship.

The girls were still buzzing from the experience of their first day on their first cruise. (“Is this a ship or a hotel?” Elise had said. “It’s a ship — with elevators!” Amelia added.)

The day had been full: lugging our bags around as I waited for our state room (I should have taken Disney’s repeated recommendations to check them); shepherding three girls under the age of 10 around a tour of the ship; practicing the requisite race to the life boats; watching the girls dance to Call Me, Maybe and Gangnam Style on a main-deck party as we cast off; finishing with an afternoon that faded to darkness on the sea, splashing in the pool.

We were all a little surprised by a kids menu that was highlighted by surf-and-turf (steak and shrimp), one that rotated every night with a choice of thoughtful, signature dishes for the children (as well as the customary chicken fingers, pizza, burgers and hotdogs for the pickier palates).

When the meal came to the table, I made a move to begin that requisite chore for any parent going to dinner with his kids: to cut everyone’s steak before you take your first bite. But before I even took my napkin off my lap, our servers Beata Nagyova, from Slovakia, and Balwan Singh, from New Delhi, had already grabbed the girls’ cutlery and began cutting the two young girls’ steaks into bite-size squares.