The experienced businesswoman will undergo a six-month notice period with Telstra before serving as Tesla Chair on a full-time basis.
Twenty-First Century Fox Chief Executive, James Murdoch, who also sits on the Tesla board, was tipped to succeed Musk before today's announcement was made.
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. In media interviews he said he often slept on a sofa in the Tesla office and more recently in an online video he briefly smoked marijuana.
Tesla CEO Elon Musk speaks at a press conference at SpaceX headquarters on September 17, 2018 in Hawthorne, Cal.
Denholm is now chief financial officer at Australian telecoms operator Telstra Corp Ltd and has been an independent director on Tesla's nine-member board since 2014.
"Robyn has extensive experience in both the tech and auto industries and she has made significant contributions as a Tesla Board member over the past four years", Musk said.
Denholm is now chief financial officer at Australian telecoms operator Telstra Corp Ltd and has been an independent director on Tesla's board since 2014.
Aside from her work at Telstra, she has also spent time at technology majors including Juniper Networks and Sun Microsystems.
"I look forward to working even more closely with Robyn as we continue accelerating the advent of sustainable energy", Musk added.
"We are undertaking a search for a new chief financial officer and will provide an update on a replacement appointment as soon as possible", Mr Penn said. The deal required Musk to not serve as chairman for three years.
Directors will create a permanent committee to ensure implementation of the terms of the SEC settlement, which include setting up procedures and controls to oversee Musk's communications - including his tweets.
Email messages sent to Denholm Thursday were not immediately returned.
That drew attention from the SEC because the tweet pushed up Tesla's stock price and harmed short-sellers who bet that the share price would fall.