Mr Musk is suing the company he helped found, alleging it has abandoned its original mission of helping humanity.
He says it is focusing on generating profits for partner and major investor Microsoft instead.
But OpenAI says Mr Musk previously backed the idea of a for-profit structure and even suggested a merger with his car firm Tesla.
In a blog post on its website, OpenAI also claims Mr Musk at one point wanted "absolute control" of the company, before eventually leaving it in 2018.
It says it intends to move to dismiss claims in Mr Musk's recently filed lawsuit.
OpenAI also published a trove of Mr Musk's emails to his fellow co-founders to dispute some of his claims.
BBC News has approached Mr Musk for comment. But he has already responded to the blog post on X, formerly Twitter.
"Change your name," he wrote.
Founding mission
Mr Musk's lawsuit, filed in a San Francisco court on Thursday 29 February, claimed that OpenAI breached principles he agreed to when helping co-found the company in 2015.
It also took aim at Microsoft.
Microsoft has invested billions of dollars in the AI firm since it established a for-profit arm in 2019, which allowed it to raise money.
His lawyers argued in court filings the firm had since become a "closed-source de facto subsidiary of the largest technology company in the world".
"We're sad that it's come to this with someone whom we've deeply admired - someone who inspired us to aim higher, then told us we would fail, started a competitor, and then sued us when we started making meaningful progress towards OpenAI's mission without him," OpenAI said in its blog.
It was written by fellow co-founders and executives including chief executive Sam Altman, Greg Brockman and former chief scientist Ilya Sutskever.
Mr Musk's lawsuit had targeted both Mr Altman and Mr Brockman, alongside the firm itself.
The blog goes into detail to dispute a number of claims made in the lawsuit.
It says that in early 2018 Mr Musk had forwarded an email saying OpenAI should join up with his electric car company, Tesla, to use it as its "cash cow".
"As we discussed a for-profit structure in order to further the mission, Elon wanted us to merge with Tesla or he wanted full control," the blog post adds.
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