Apart from Agarwal, Musk is said to have sacked the company's chief financial officer Ned Segal and head of legal policy, trust, and safety, Vijaya Gadde, the Washington Post reported, quoting sources.
Agarwal and Segal also left the platform's San Francisco headquarters, and will not be returning, CNBC reported.
The report came hours before the court's deadline for Musk to take over the company, and thus make good on the deal he had made to buy the platform.
'The Bird Is Freed': Musk
He also put out a tweet on Friday, saying, "The bird is freed," in an apparent reference to the logo of Twitter.
On Thursday, Musk had tweeted that he was buying Twitter "because it is important to the future of civilisation to have a common digital town square, where a wide range of beliefs can be debated in a healthy manner".
He had also visited the Twitter headquarters with a kitchen sink in his hands, and changed his bio on the platform to 'Chief Twit'.
The deal to acquire Twitter, worth $44 billion, had started in April this year, when Musk announced that he had a 9.2 percent stake in the company, making him the biggest shareholder.
However, in May, the Tesla boss said that he no longer wanted to acquire the company over concerns regarding the number of fake accounts and 'bots' on the platform. However, Twitter argued in court that Musk was legally obligated to go through with the acquisition, which led to a months-long trial.
(With inputs from The Washington Post.)