The first work visitors see at the imposing new National Gallery Singapore — a $380 million complex housed in the former Supreme Court and City Hall — is a small 1865 print that could almost be a parody of colonial art.

The image, by the German illustrator Heinrich Leutemann, shows a tiger leaping out of the jungle at George Drumgoole Coleman, the Irish architect who helped lay out the city in its early days as a British colony. Coleman stands his ground, looking startled, while his turbaned, dark-skinned servants fall down.

“We looked high and low to see if this actually happened, and we don’t think it did,” said Shabbir Hussain Mustafa, a senior curator at the gallery, which wrapped up two weeks of opening celebrations on Dec. 6. He pointed to another historic picture, of a bustling port. “It’s supposed to be Singapore, but it’s got camels and minarets.”

Those early works, with their blend of myth and reality, colonialism and modernization, set the tone for the National Gallery’s collection, which traces the cultural and artistic shifts in Southeast Asia from the mid-19th century to the present day, including events like the Vietnam War and the effects of rapid industrialization on an agrarian society.

“This is the first major attempt by an institution to put together a collection of modern and contemporary art from Southeast Asia,” said Claire Hsu, executive director of the Asia Art Archive in Hong Kong. “It hasn’t been done before at this scale or with this depth.”

Singapore was long seen as a city equally focused on commerce and on controlling its residents’ habits (it once banned chewing gum), a paternalistic view that included keeping an eye on the entertainment and culture its citizens were consuming.

A decade ago, Prime Minister Lee Hsien Loong, the son of Singapore’s founding father, Lee Kuan Yew, who died in March at 91, mapped out a vision for the continued renewal of its civic district that would help Singapore overcome its reputation as a cultural desert. With trademark efficiency backed by hundreds of millions of state dollars, cultural landmarks were opened or renovated one by one: The National Museum of Singapore in 2006; the Peranakan Museum of local culture in 2008; and the Victoria Theater in 2014.

“The reality is that there will be a national agenda, to put itself in the center of the region,” said Ms. Hsu, whose Asia Art Archive has collected about 50,000 documents. “My hope is that the gallery will be a gatekeeper of these works, not merely a keeper. It should not own history, but inspire and enable others to engage with it.”

More than 600 million people live in the 10 Asian nations roughly south of China and east of India, in what Eugene Tan, the director of the National Gallery, described as “one of the most culturally diverse regions in the world.” Yet much of its art has been overlooked by the West, and many significant works have been confined to private collections, partly a legacy of colonialism.

Singapore started collecting Southeast Asian art soon after becoming an independent state in 1965, which it achieved after British colonialism, Japanese wartime occupation and an uneasy two-year stint as part of Malaysia. But until now, this part of the national collection did not have a permanent exhibition space.

To create the National Gallery, the old Supreme Court and City Hall buildings were joined by a vaulted glass atrium held up by giant metal trees. Fashionable restaurants line a rooftop garden, while the buildings’ interiors have been left intact: pillars, marble floors, high teak ceilings and walls painted blood red.

A domed library is now an art research center, and two empty prison cells stand to remind visitors of the building’s past. (The museum’s promotional video starts with the ominous line: “Criminal masterminds were once sentenced to hang here.”) The “surrender chamber” in City Hall, where Japanese capitulated on Sept. 12, 1945, and where Lee Kuan Yew was sworn in as the first prime minister in 1959, has been artfully preserved.

The museum has opened with about 1,000 objects on display, divided between two exhibitions. “Between Declarations and Dreams” covers the broader region, while “Siapa Nama Kamu” (which means “What’s your name?” in Malay, one of Singapore’s four official languages, along with English, Chinese and Tamil) is focused on the city-state.

One of the highlights of the first is “Forest Fire” (1849), the largest known painting by Raden Saleh, a pioneering artist from what was then the Dutch East Indies (now Indonesia). A bull and tigers tumble ferociously over each other in a large oil that was originally a gift to King William III of the Netherlands.

“Because of the area’s colonial history, we had to cast our eyes beyond this region,” Lisa Horikawa, another senior curator at the gallery, said of the difficulties of retrieving works like “Forest Fire” from private collections. She added that “Forest Fire” required conservation, and that it took three years to bring it to Singapore from Europe.

Indeed, the gallery relied heavily on loans for its inaugural shows. One high-profile loan is “Put Down Your Whip,” an oil that the Chinese artist Xu Beihong painted in Singapore in 1939, on the cusp of World War II in the Pacific. It shows the actress Wang Ying as a young woman who had escaped Japanese occupation. The painting disappeared from public view in the 1950s and only re-emerged in 2007, when it was acquired by an anonymous buyer for $9.2 million at Sotheby’s in Hong Kong, a record high price for a Chinese painting at the time.

Many works in the collection reveal cross-cultural influences between the West and Southeast Asia. Some are unabashedly pro-colonialist, like “Spain and the Philippines” (1884) by Juan Luna, a Filipino artist who exhibited his work in Europe. It depicts two women in draped robes, with “Spain” leading the “Philippines” up a flight of flower-strewn stairs into, presumably, the golden dawn of Western enlightenment.

Vietnamese works combine French Impressionist influences and traditional materials like silk and lacquer. Landscapes by the Indonesian artist Wakidi evoke the glowing pastel tones of Monet.

Also of note is a small 1940s self-portrait by Georgette Chen, a cosmopolitan Singaporean painter who grew up in Paris and New York before returning to Asia. “Singapore was inherently global, long before we talked about global art,” said Mr. Mustafa, the senior curator.

Starting in the 1950s, the tone becomes darker, as local artists focus on social turmoil. “For the younger generation of artists, the romanticized image of the region was not sufficient,” he said.

Two works important to the city-state’s identity are “Epic Poem of Malaya” (1955) and “National Language Class” (1959) by the Singaporean artist Chua Mia Tee. Both show a local teacher or speaker inspiring groups of youth. Such images marked “the end of colonialism and the beginnings of social realism,” said Mr. Tan, the gallery’s director.

A gallery that focuses on the Vietnam War has a wall of 17 illustrations by artists sent into the field by the North Vietnamese Army. Facing them is Le Huy Tiep’s “Blood of War,” from 1972, in which corpses lie at the feet of a blindfolded man, while a mother clutches a naked child. The tips of machine guns are visible at the edge of the frame.

Over the next year, the National Gallery Singapore will be working with overseas institutions to put Southeast Asian art into a global context. Chinese museums will contribute to an exhibit space for the artist Wu Guanzhong. The Centre Georges Pompidou in Paris will lend about 100 works to a joint exhibition next spring, and “Arts and Empire,” a retrospective of British colonialism at Tate Britain, will move to Singapore next October.

Mr. Tan said the rise of a new generation in Singapore, less focused on the city-state’s emergence and eager to be part of the larger world, added urgency to displaying art from the city-state and its surrounding region.

“I think I only went to one museum as a kid,” he said, “and that was on a school trip. Singapore is a very different place now. And having a permanent space for our collection is a game changer.”