India have won the match
is not a grammatical mistake. We're not talking about the nation, India per se, what we have here is an example of metonymy, when the name of one thing substitutes the thing itself. In this case, India substitutes the (cricket) team representing India. A famous example of metonymy is "Washington" which we understand it to mean the President of the USA, or its executive office. Likewise, in the UK it is common to refer to the British Prime Minister or the government as "Downing Street" as in:
Downing Street has rejected claims that David Cameron described
environmental levies as “green crap” as the coalition explores ways to
minimise the impact of green subsidies on household energy bills.
In the OP's phrase it is clear that India refers to a team, let us suppose it is the national cricket team. A cricket team is made up of eleven players, a player is countable thus, in English, team is a collective noun, which can be considered either singular or plural. When we are considering India as a single unit we use the singular verb, when we think of the individuals who make up the team (who play for India) we use a plural verb. An article from New Zealand Herald, January 16, 2014.
India are the world's No 1 ODI side, New Zealand are eighth.
What the journalist is really saying is that the cricket players who play for the Indian team are the world's NO 1 One Day International side players.
Grammar Monster has this to say about collective nouns, which I believe sums it up nicely.
A collective noun can be considered as either singular or plural
depending on the sense of the sentence. If it's too hard to make a
decision on singular or plural, precede your collective noun with
words like members of…, forcing you to go plural.
What I have described above is especially true for British, Australian and New Zealand English. However, in American English, the singular verb is usually preferred. Below is an example taken from:
The Washington Times Communities 1
It is the fourth time the U.S. has defeated Mexico with a 2-0
scoreline in Columbus.
Mignon Fogarty AKA Grammar Girl says this on the matter
Americans tend to treat collective nouns as single units, so it’s more
common to use the singular verb unless you’re definitely talking about
individuals (3). So in America you would be more likely to hear “The
faculty is meeting today” than “The faculty are meeting today.”
Many thanks to @snailplane for giving me the heads up on this one