The Jacksons Landing site has been disused since 2004 and was purchased by Hartlepool Borough Council in 2013.
The funding was granted at the council's meeting to approve the 2016/17 budget on Thursday.
Critics have said there are still no firm plans for the site and demolishing the mall is a waste of money and that it should be redeveloped.
The council said the demolition, which was approved in July, would save the cost of maintaining the building and could help with wider regeneration.
It added that there were developers interested in the site and the area but none of them wanted the mall.
Councillor Paul Thompson, who is opposed to the demolition, said: "Nobody is saying this area shouldn't be redeveloped. What people are objecting about at the minute, is pulling it down without any plan.
"If we pull it down we limit the options we can put on here. A developer may come along and say they'll develop the existing site, that would be great because we've just saved the taxpayers of Hartlepool £40,000."
The council bought the former retail unit at the town's marina for £1.5m in 2013 after it stood empty since 2004.
The five-acre site was previously owned by investment company Schroders.
Demolition is expected to start in the next few weeks.

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Ryan Farquhar took victory in the Open race, run on Friday night, the classic race and the Moto 450 outing.
William produced a dominant ride in the main event of the meeting, taking the chequered flag by 28 seconds from runner-up Jamie Hamilton.
Mullingar's Derek McGee was third on the Wilson Craig 600cc Honda.
Earlier, William followed up his double win in the 600cc category at Cookstown by winning the Supersport class from McGee and Hamilton in difficult conditions.
Weather conditions were better on Friday evening when Farquhar won a thrilling Open race by 0.4 seconds from Tyco BMW pilot Dunlop.
Early leader Hamilton, from Ballyclare, completed the podium positions.
McGee took the flag in the Supertwins class, seeing off the challenge of Hamilton.
Jeremy McWilliams, making his debut over the 5.3-mile circuit, was third in the Classic race.
Funding for the demolition of a shopping mall owned by a council has been approved at a cost of £40,000.
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William Dunlop clinched a double success at the Tandragee 100 meeting, winning the feature Superbike race and the Supersport 600cc event.