"He is an ambitious young manager who is enthusiastic about the game," said Foxes chairman Milan Mandaric.
"I needed someone with hunger, drive, commitment and potential to aspire to all our dreams and aspirations. And I believe Martin ticked all the boxes.
"He was unquestionably the outstanding candidate to come before me and deserves the immense opportunity that awaits him."
Caretaker manager Nigel Worthington and Neil Warnock had also been shortlisted to take over at the Walkers Stadium.
Worthington took charge of the side in April, when previous manager Rob Kelly was sacked.
Allen has not managed at Championship level before, but spent two years at Brentford before quitting last summer in a row over money and heading to the National Hockey Stadium.