The retailer will launch the initiative to customers later this month with a prime-time TV advertisement featuring a host of celebrities, including Martine McCutcheon, Paul Daniels, Frankie Dettori, Alan Titchmarsh, Ronnie Corbett and Alan Whicker.
Checkout staff are to ask customers how many bags they are reusing and agree on the number of points to be awarded. Larger bags such as reusable trolley bags may earn more than one point.
Clubcard statements mailed to customers will in future include a separate box for 'green' points.
Tesco said the bags initiative was the start of a wider plan to use Clubcard points to encourage customers to adopt more environmentally friendly shopping habits.
CEO Sir Terry Leahy said: "We have had a team looking at carrier bags, trialling different ideas in our stores and talking to customers about what we could do to encourage them to use fewer bags and to recycle the ones they do use. The results of our research tell us that we have to move the emphasis away from trying to force change and on to rewarding positive behaviour. In other words, a more carrot than stick approach."
The scheme goes nationwide on Monday, 14 August, with the TV advertisement due to air on the same day.
Under its ten-point community plan, published in May, Tesco has pledged to cut the number of bags it gives out by 25% over two years, which will equate to one billion fewer bags every year. All carrier bags given out by the retailer will be biodegradeable by this September.
Tesco said it had also introduced bigger, thicker bags into many stores so that customers can get more of their shopping into fewer bags. Online customers are also being urged to give their plastic bags back to delivery drivers for recycling.
Over the next 12 months, every Clubcard customer will be given a coupon entitling them to a free reusable Bag for Life.