Part of it is going to be that the region (HK, Singapore, SK?, China) was hit much harder during the 2003 SARS epidemic so spent more time getting prepared.
Also, it has long been the expectation that the next epidemic would come out of China, both due to population density and proximity to livestock/wild animals, so those countries could expect another "gift" from their Chinese neighbors.
Japan? It's relatively unscathed so far, but it is an isolated island, with comparatively few foreigners which isn't on great terms with its Asian neighbors. As @bobsburner mentioned, "travel density" counts for a great deal with covid-19. For now - counting it as luck, might become skill later. Or its success might just be down to not-testing, as @steros says.
Supply chains are in Asia. This is being shown as an Achilles' heel of Western service-oriented economies: if we need to ramp production of electronic testing kits or ventilators, we are cut off from a lot of the world's manufacturing capacity.
Collectivism vs. individualism has a part to play in it as well: it's more difficult for a Western politician to push drastic measures like quarantine (let alone China's dictatorial system). Still, Italy was pretty quick to do so, because they had no choice. Still, the cultural traits leading to quick acceptance by the community of harsh limitations on individual liberties seems to be an Asian advantage at this point:
Here's a Medical Sciences Stack Exchange answer re. isolation, testing and surveillance procedures followed in Guangdong, near HK, so not Wuhan at all. An article partly covering Wuhan itself. And we've heard of the one-week hospitals. We're just not there yet, in terms of mobilization. Or coercion.
Constant deficit spending. A country like Italy is at 130%+ GDP debt. They've resisted pressure to shape up their finances, tax collection, pension payments, etc... for decades. They just don't have anything left to surge spending in emergencies. This applies to just too many European economies.
It's also really too early to say much about the final outcome. Currently South Korea and China are better at getting the disease under control, but what really counts is the long run. If a few dozen people in Wuhan spread the disease rapidly in January then we need to be ready to wait this out for a while - "social distancing & quarantines" for 3 months, then relaxing will result in restarted epidemics.
There's also, esp. at the start, plain luck. The 2003 SARS epidemic hit Canada's west coast less than Toronto, because a doctor got bad vibes about a patient with breathing problems and she isolated them, before any outbreak was known. In Toronto the same type of cases mingled with the general hospital staff and patients and the outbreak was much worse. In Italy 1 covid case was not identified as such and went to the hospital 4 times, interacting with everyone else. S. Korea had a massive cluster early on, but it was within a religious sect, which might interact less with others.
It's hard to really assign a "good vs bad" label to early spread. Past that, the quality of political leaders and their ability to balance out economic/social reality vs medical advice (which is still operating in uncertainty - witness the UK's "herd" approach) is going to be key. This is a good time to be listening to experts, and there some of the Asian technocrats are a definite better bet than some of the West's populists. I know I wouldn't be thrilled to have Duterte in the Philippines in charge, for example, and he's Asian.
Rather than competition and comparisons, it might be useful to see what has been achieved so far: to some extent S. Korea and China have shown that aggressive testing and quarantines can slow the disease, Australia was very quick to sequence the bug's genome, we got a test within weeks! and some Western countries are starting to look at and evaluate candidate vaccines. We are much better off in 2020 than if SARS had had that kind of punch in 2003.