The company's share price rose by 2.87 per cent in late morning trading on the news.
Murphy is stepping down to "pursue other opportunities", with David Lockwood, who has experience in defence, taking over.
The company said: "The board thanks Bob for his leadership during challenging times, including: the development of a clear strategy with a focus on connectivity; driving technology investment resulting in a number of strategic wins for the group; and enhancing the company’s customer focus and performance to customer expectations. We wish him well in his future endeavours."
Lockwood will join no later than 1 January 2017 and will join the board.
Laird is a FTSE 250 technology providing products and solutions that protect electronics from electromagnetic interference and heat and which enable connectivity in mission-critical wireless applications.
"David has built the company into a focused enterprise connectivity company with leading positions in the connected transport and consumer device markets, delivering sustained organic growth," Cobham said.
Lockwood though was also formerly vice president of global defence and security at BT Global Services.
Cobham chairman John Devaney said, “David has a background in defence and communications technology which are at the heart of Cobham. We look forward to him joining the company and thank Bob for his contribution.”
In June Cobham poached new finance boss David Mellors from QinetiQ in a bid to improve the firm's fortunes.