Dinsmore, aged 44, has been director of operations at News International since October, having worked his first shift on The Sun at the age of 22.
He became editor of the Scottish edition in 2006 and was promoted to general manager of the company's operations in Scotland in 2010 followed by a brief stint as managing editor in London.
Mohan is leaving the paper to take up a senior role advising chief executive Mike Darcey for the new company across Europe following the official formation of new News Corp, which will be incorporated next Friday (28 June).
Mohan, who has been with the Sun for 17 years and edited the paper since 2009, will be based in London but will report directly to Robert Thomson in New York.
Dinsmore, who takes control of the tabloid on 24 June, said: "There is no better job in journalism than editing Britain's most popular paper.
"I relish the opportunity to build on the historic strengths of the Sun, and harness new digital opportunities to offer our loyal readers more than ever."
Darcey said: "These are exciting times for our company, with new leadership on all our titles and an Executive team that has been transformed in recent years.
"We will join the new News Corp next week in a strong, determined and confident position and I have no doubt that we will secure a sustainable future for the world-class journalism we all cherish."