They were Super Rugby champions in 2011, third in 2012 and fifth in 2013.
Graham would now report to a general manager of the high performance programme, former Wallabies centre Daniel Herbert, the Queensland Rugby Union said in a statement.
"We fully understand that the results of the Reds over the last two seasons have been unacceptable," QRU chairman Rod McCall said.
"However, we feel strongly that Richard is the right man to lead us into the 2016 season under this new structure."
McCall said more than 20 people had applied for the head coaching job and Graham said he was aware that any further failures would not be acceptable.
"I am humbled by the opportunity to coach this team again in 2016," Graham said.
"As a squad we will not tolerate any excuses that compromise our preparations or performance in order to achieve success in 2016."
Graham, who will work alongside former All Blacks' lock Brad Thorn and Nick Stiles next season, will need to replace Will Genia and lock James Horwill, two lynchpins of the 2011 triumph, as well as former New Zealand loose forward Adam Thomson.
Flyhalf Quade Cooper's future is also reportedly still in limbo after he signed a new contract with the Australian Rugby Union and pulled out of a lucrative contract with French club Toulon, who have threatened to sue.
(Reporting by Greg Stutchbury in Wellington; Editing by John O'Brien)
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