Although Italian non-industrial craftsmanship was the strongest economic force in Western Europe prior to their overtake by industrialized nations in the first half of the 19th century (see the research by Maddison, updated by Bolt and Van Zanden in 2013) the decision to promote "Made In Italy" on the world stage was a concentrated effort by the Italian Government headed by Bettino Craxi in the 1980s, flanked by the efforts of administrators of key Italian cities like Milan and Turin, run by members of Craxi's Socialist Party.
The Italian Economy never truly fully industrialized. In spite of centralized control of key industries through the state-run holding company, FRI, by the Fascist regime which continued in the post-war era, most Italian companies remained small. A holdover from the country's long history of political fragmentation, the Italian economy was dominated by many fragmented family-run companies. Large industrial conglomerates were few and far between, and mostly concentrated in the cities of Turin, Milan, and Genoa. This meant that Italians were bringing fabric to tailors to make clothes well into the 1950s and 1960s, when their English and American counterparts were shopping in department stores.
However, the the experience of isolationism (and war) meant that post-war Italian governments were set on international integration; borders were opened, and trade barriers were lowered. Italy became one of the strongest sponsors of the European Integration process. With cheap industrial products coming in from abroad dramatically changing Italian society in the 1970s, small Italian companies had to find way to adapt, and the adaption came through quality. They couldn't beat foreign imports on prices, but they could beat them by justifying their higher prices.
By the 1980s, traditional artisans with a good sense for business had been able to build on the heritage of their forefathers and break out onto the international scene, compensating lost domestic consumption with foreign exports; Aldo Gucci, Gianni Versace, Giorgio Armani, and others became known around the world for the quality of their work. Workshops quickly became factories; marketing managers were hired, and the Italian fashion industry was born. Beyond fashion, this hyper-localized products like foodstuffs, and even automobiles or firearms, followed the strategy of maximizing local heritage and traditions to create unique, high-qualify products. Consortiums guaranteeing the quality of products emerged, as producers realized the importance of reputation in ensuring consumers were willing to pay for higher prices. These consortiums are still vital today in ensuring that Italian goods are made to certain quality standards.
The Italian governments of the 1980s took advantage of this shift in strategy by traditional Italian companies. Helped along by massive public spending fueling demand, Italian companies were incentivized to internationalize, while a renewed interest in European integration (which had stalled in the 1970s) increased the ties between Italy and the rest of Europe. Piedmont and Lombardy became post-industrial tertiary economies, while industry in the Veneto, Emilia-Romagna, Tuscany, and Marche went from being emergent in the 1970s to an economic force in the 1980s, adding more small, quality Italian firms to the international sphere. In the 1980s, Craxi's government distributed largesse to his fellow socialists in the Milanese and Turinese city administrations, allowing them to stage elaborate international conferences and events. Even the Socialist Party congress in 1988 included a pavilion exhibiting goods that were "Made in Italy."
To this day, Italian consumption patters stay true to the country's historic aversion to industrialization; Italians buy Italian washing machines, because they have specific wash cycles gentle enough for expensive Italian clothes. Italians don't like to consume very many goods, but those that they do consume tend to be very expensive. Italians are extremely specific about the way they cook; restaurant culture paled in comparison to the culture around produce markets until celebrity chefs started appearing on TV, prior to that restaurants were for unmarried urban professionals and foreigners.
Sources that could interest you:
The Italian history channel ran a very interesting series on the 1980s, including an episode on Milan, the epicenter of the Italian export boom (in Italian).
Ginsborg, Paul. A History of Contemporary Italy, 1943-1988. London: Penguin.
Gilbert, Mark. The Italian Revolution: The End of Politics, Italian Style? Boulder: Westview, 1995 (especially the early chapters give a very good overview of the Italian economic scene in the 1980s; the rest of the book explains how it all came crashing down).