Musk said in August he was considering taking Tesla private in a $420 a share and that he had secured funding for a deal that was later scuttled but attracted scrutiny from several government agencies.
Two days after the lawsuit, Musk reached a settlement with the commission, agreeing to a personal fine of $20 million, a separate $20 million fine by Tesla and his resignation as chair.
An Australian, Denholm is Telstra's chief financial officer and head of strategy and already has served on Tesla's board as an independent director.
She will give up these positions after serving a six-month notice period to take up a full-time chairman position at Tesla.
The carmaker last month quieted some critics after it reported a net profit and positive cash flow in the third quarter as higher production volumes of its crucial Model 3 sedan began to pay off, delivering on Musk's promise to turn the company profitable.
Tesla has endured a hard few months, following comments Mr Musk made on social media in early August suggesting he had "funding secured" for a deal to take Tesla private.More news: Anthony Joshua reveals how Tony Bellew can defeat Oleksandr Usyk
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Musk cited Denholm's long experience in the tech and auto industries as strong advantages and said in a statement that he looked forward to working with her. In the meantime, she'll work closely with Musk.
Robyn Denholm has been an independent director on Tesla's nine-member board since 2014.
Besides Telstra, Denholm has worked in other tech companies such as Juniper Networks and Sun Microsystems, and she also worked for Toyota, the statement added.
"To ensure a smooth transition during the remainder of Robyn's time at Telstra, Elon will be a resource to Robyn and provide any support that she requests in her role as chair", the statement from the electric car-maker said.
In the settlement, the SEC pulled back from its demand that Musk, who is synonymous with the Tesla brand, be barred from running Tesla, a sanction that many investors said would be disastrous.
Neither Musk nor Tesla admitted or denied the SEC's findings as part of the settlement.