Leahy will be replaced by Philip Clarke, who currently has responsibility for the group's international operations in Asia and Europe.
Tim Mason, the current US chief executive and former marketing director, will keep his current role, as well as becoming group deputy chief executive with responsibility for branding, company values and climate change.
Brasher is to become chief executive of the UK business with responsibility for the Republic of Ireland.
Leahy said he was preparing to retire at the age of 55, feeling his work is "almost complete".
"When I became CEO I had a plan to build Tesco around its customers, to make it number one in the UK and to find new long-term growth in non-food, in services and in international expansion.
"I wanted to develop a purpose and values that could sustain Tesco through its challenges and encourage and grow future leaders."
"It has taken 14 years but that strategy has become a firm reality and so I feel my work is almost complete."
This is likely to leave the UK company needing a senior marketer, presenting the possibility of further marketing job moves in the supermarket sector following today's move by .
Brasher's promotion will kick off the recruitment process for one of the most high-profile in the country, with Tesco understood to be already engaged in a hunt for his successor in the commercial director post.
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