The weekend vote had been seen as a test of Prime Minister Narendra Modi's popularity after more than four years in office, but no clear winner had emerged after the high-stakes contest.
However, the governor of the state, who has the legal power to decide who forms the government, invited the BJP, triggering cries of protest from opposition parties.
In response to the decision, politicians from the Indian National Congress party, which had been in control of the state prior to the election, delivered a petition to the Supreme Court Thursday morning, in a bid to stop the formation of the BJP government in the state.
Opposition politicians also accused the BJP of trying to poach candidates to help form the majority they need.
"The BJP's irrational insistence that it will form a government in Karnataka, even though it clearly doesn't have the numbers, is to make a mockery of our constitution," said the Congress Party President Rahul Gandhi on Twitter.
"This morning, while the BJP celebrates its hollow victory, India will mourn the defeat of democracy," added Gandi.
The election was seen as a barometer of the public mood ahead of general elections in 2019.
If the BJP goes ahead and forms the regional government in Karnataka, the opposition Congress party will rule just two of India's 29 states and one union territory. The BJP currently rules 22 states, either by itself or as part of an alliance.
No outright majority
Under the parliamentary system, the party with the majority of seats is the leader of the new government. Yet, on Tuesday, results showed none of the three main political parties were able to secure an outright majority in the 222-seat assembly.
The BJP had the largest number of seats, with 104 out of 222. The next two largest winners were the Indian National Congress party with 78 seats, and the regional Janata Dal (Secular), also know as JD(S), with 37 seats.
While the Congress secured fewer seats than its main rival, it won a greater number of votes -- getting 13.8 million votes compared to the BJP's 13.2 million.
Now, newly sworn in chief minister, BS Yeddyurappa, will have to prove that he is able to claim the support of a majority in the state's lower house within 15 days, according to the state's governor Vajubhai Vala.
Constitutional law experts say a state governor has a lot of discretion in choosing which party to invite to form a government, whether it's the party with the highest amount of seats, though short of a majority, or a coalition of parties.
But after the invitation, the head of the invited party must show he or she has the support of a majority of the legislature's lower house.
"The constitutional position is that it's entirely up to the governor of the state to decide," said Subhash Kashyap, the former secretary general in India's lower house of Parliament.
"So far as precedent is concerned, there are all kinds of precedents, different governors have used their precedents in different ways," added Kashyap.