The 2,000-seat opera house was packed for the opening after tickets sold out months in advance, leaving many scrambling to buy them on the black market in the days preceding the event.
Domingo launched into the event with a series of performances including Rossini's "The Barber of Seville" and the Broadway musical "West Side Story".
The venue will also host local shows, with popular Emirati singer Hussain Al Jassmi performing there in October.
During rehearsals, Dubai ruler Sheikh Mohammed bin Rashid al-Maktoum visited the site where he greeted the Spanish tenor, according to a video posted on the Dubai government's Twitter account.
"Cultural development was always present among our development plans and vision for the future... We have provided the infrastructure to support it," he said in remarks published on his website.
Organisers have said that an official opening gala will be held later when work on the venue is completed.
On Tuesday, the Dubai Opera management organised a media tour as workers struggled to prepare the theatre for Domingo's performance.
Signup to our newsletter and follow us on Facebook and Twitter!
The opera sits at the foot of Burj Khalifa, the world's tallest tower, which was built by the same developer, Emaar Properties.
In a nod to Dubai's long history as a port city, the opera house is shaped like a dhow, a traditional wooden boat used for centuries in Gulf waters.
But the ultra-modern venue can transform into three modes, operating as a theatre, concert hall and a flat-floored hall suitable for banquets and weddings.
In the space of decades, Dubai has transformed itself into a centre for trade, travel and tourism.
Spending trillions of dollars earned from oil exports, it put itself on the map with luxury resorts, glitzy skyscrapers and artificial islands shaped as palm trees and a world map.
But its cultural scene remained low profile in the business-oriented emirate.
Jasper Hope, chief executive of Dubai Opera, said the opening represents "the start of a new journey for arts and culture in Dubai".
Opera remains very much a cultural import for Gulf Arabs -- the only other opera house in the region is in the Omani capital Muscat.
But Dubai's population is predominantly foreign, including a sizable Western community.