Scotland lost 3-0 to Kazakhstan in their opening qualifier and could only beat San Marino, the world's lowest-ranked team, 2-0 last month.
“It was only taken after full consideration by the Board and after an honest and respectful conversation between myself and Alex earlier in the week," Scottish FA chief executive Ian Maxwell said in a statement.
"Ultimately, the performances and results as a whole in the past year - and, in particular, the manner of the defeat in Kazakhstan – did not indicate the progress expected with a squad we believe to be capable of achieving more.
“With that in mind, it was agreed that we would seek to recruit a new management team to take us forward for the remainder of European Qualifying campaign."
McLeish's assistant coaches, Peter Grant, James McFadden and Stevie Woods, will also leave their roles.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
