The Under-20 boss replaces Gareth Southgate, who will look after the senior team for their next four matches after the departure of Sam Allardyce.
A point in Kazakhstan on 6 October will be enough for the Under-21s to secure a spot in next year's finals in Poland.
Their final group match is at home to Bosnia and Herzegovina.
Boothroyd's assistant coach Paul Williams will take charge of the Under-20s on an interim basis.
Boothroyd has coached the England Under-20 side since 2014.
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Former Watford and Coventry manager Aidy Boothroyd will take charge of the England Under-21 team for their final two Euro 2017 qualifiers.