The landmark decision follows a five-year battle by the company to build a store in the town, the only postcode in mainland Britain without a Tesco superstore.
The proposed site is at a former gas works at Skipton Road passed through the planning committee with 11 councillors out of 13 voting in favour.
The council approved the application but it must still be referred to the government for final approval.
Tesco withdrew an application in 2007 after objections – but submitted the current one in 2009.
The initial objections were about traffic and impact on local businesses, but Tesco revised plan is for a smaller and more eco-friendly store, which would create 360 full and part-time jobs.
Preliminary work will include highways widening and strengthening of the Skipton Road bridge.
Tesco’s regional corporate affairs manager, Matthew Magee said: Obviously we are pleased with the decision from Councillors. I think all along we have said it is a popular application and it was good to hear that come from Councillors directly.
“I think highways was always going to be the issue and again it comes down to the professionals making a judgement that highways will be acceptable around the site and that the wider network will improve as a result of the store.”