Mr McLeish was effectively deputy to the late leader Donald Dewar but without the official title.
Like his leadership opponent, Jack McConnell, Mr McLeish said the party found itself in the unfortunate position of having to choose a leader after the untimely death of Mr Dewar.
However, he promised that if elected he would carry Mr Dewar's "vision" forward.
He said: "The tragic and untimely death of Donald Dewar has created a set of circumstances and a contest no-one wanted.
"Following discussions with colleagues here in Scotland and widespread representations from the Scottish Parliamentary Labour Party, the party's Scottish Executive and members of the Scottish Government, I am honoured to put my name forward as Labour Leader in Scotland.
"After 300 years, under Donald's leadership the parliament was delivered.
"I was his deputy through it all. Having been at his side, leading the Constitutional Steering Group and building the 'new politics', I know how vital it is to carry forward Donald's vision.
"It is my determination to lead an open and inclusive government, to form a bond of trust with the people of Scotland and to make integrity our watchword.
"Much has been said in the recent sad days about 'unfinished business'.
"The existence of the Scottish Parliament has never been an end in itself, but the means to achieving our ideals.
"We are already making great strides towards ending child poverty, creating opportunity for all and jobs for all and these will be the hallmarks of my administration.
"It goes without saying that only a Labour government can deliver these and I am confident I can lead that government to electoral success.
"I welcome the fact the Labour Party's democratic process will be seen by all to be fair and I hope the contest will be completed with dignity."