The 62-year-old Scot, who worked as first-team coach at Liverpool under Rafael Benitez from 2004 to 2008, replaces Dmitry Radyukin, who was sacked on Monday.
"The negotiations had taken a long time to complete," Sibir chairman Stavislav Zhuravsky told the club's website.
"We had five candidates to chose from but decided on Miller because he had worked at a top club in the best league in the world. We know he is a top professional and has a vast scouting experience."
The much-travelled Miller has also coached Japanese club JEF United Chiba and Swedish side AIK among others.
"The results and performances of the team under Radyukin had been disappointing," added the Sibir boss. "We had expected a lot more."
Ambitious Sibir are eyeing a quick return to Russia's top flight after playing there in 2010. They are in fifth place in the first division with the season resuming in late March.
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