Asda also committed to reducing waste by 50% and having a net "regenerative impact on nature" no later than 2050. It said its customers will not pay more for greener options.
The retailer's move responds to growing consumer demands for less waste and more action on the environment.
Britons have become increasingly aware of the urgency of addressing climate change, spurred on by campaigners including veteran naturalist and broadcaster David Attenborough and Swedish teen activist Greta Thunberg.
"This is an issue that matters greatly to our customers. Our own insight tells us that more than 80% believe that supermarkets have a responsibility to reduce the amount of single use plastics in stores," said Asda CEO Roger Burnley.
In January, Sainsbury's committed to having net zero greenhouse gas emissions by 2040, a decade ahead of the UK government's own target which it said "isn't soon enough"
Earlier this month, American parent Walmart agreed to sell a majority stake in Asda to the Issa brothers and private equity group TDR Capital in a deal which gave the British business an enterprise value of $8.8 billion.
The British billionaire brothers made their fortune from petrol forecourts.
To coincide with its pledges, Asda launched its first sustainability trial store in Leeds, northern England, which is designed to help shoppers reduce, reuse and recycle.
More than 30 household staples are sold in refillable format, including Unilever's PG Tips tea bags, Persil laundry detergent and Radox shower gel; along with Kellogg's cereals, PepsiCo's Quaker Oats; Vimto cordial and Lavazza coffee.
Some 53 fresh produce lines are sold in a loose and unwrapped format including 29 new lines such as cauliflowers, mushrooms, apples, cabbages and baby plum tomatoes.
The outer plastic wrapping on Heinz and Asda brand canned multipacks including beans and soups will be removed.
The store will also feature a reverse vending machine for cans, plastic and glass drinks bottles and a hanger recycling facility.
(Reporting by Molly Darlington and James Davey. Editing by Jane Merriman)