The firm is bulldozing the 1960s Trinity Square complex to make way for a supermarket, apartments and a cinema.
The stark concrete building, made famous in the 1970s crime film starring Sir Michael Caine, was originally due to be knocked down in autumn 2007.
But demolition was delayed to allow the development's market hall to continue trading throughout Christmas.
The hall was vacated on Monday, and Gateshead Council said it was working with any businesses or jobseekers in need of funding, premises, training or employment.
Tesco's development arm, Spenhill, will now begin work on the multi-million pound scheme.
An exact date for the car park's demolition is expected in the next few weeks.
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