Clubcard TV, offering content from Warner Bros, Endemol, Aardman and others, is supported by tailored adverting based on Clubcard Data.
After rolling out in beta in February, the service went fully live online today. Tesco hopes to expand it to other platforms, such as games consoles and tablets, in the future.
Clubcard TV has been developed by the team behind Blinkbox, the movie and TV download platform that Tesco acquired in 2011.
The 16m Clubcard customers in the UK can sign up to the service with their existing Clubcard details, including their 16-digit member number, date of birth and postcode.
They can then access a range of movies, such as Brad Pitt western The Assassination of Jesse James by the Coward Robert Ford (Warner Bros) and The Shawshank Redemption (Daro Media), along with TV shows including The Only Way is Essex (All3media), Doc Martin and The Real Hustle (DRG).
Children's content includes movie The Wind in the Willows (DCD Rights) starring Matt Lucas and TV series The Care Bears and Strawberry Shortcake from distributor American Greetings.
All this content is available to watch for free, but Tesco will make money from Clubcard TV by serving up advertising to users based on data gathered about their shopping habits.
Advertisers signed up for the launch of the service include Kellogg's, J&J, Colgate and Danone.
"Clubcard TV offers a whole world of free entertainment for all the family. It makes digital entertainment easy and accessible for our customers," said Michael Comish, the co-founder of Blinkbox who is now chief executive of Tesco Digital Entertainment.
"The reason we can offer great programming for free is because customers will see relevant advertising before and during the movie or TV show they are watching.
"We'll use Clubcard data to tell us what might be relevant for our customers and therefore help us deliver a more personalised service."
Warner Bros president of International Television Distribution Jeffrey R Schlesinger added: "Tesco has stepped up and made a financial commitment enabling them to license a selection of films for an exclusive short window sequenced between the traditional free TV windows.
"We are very pleased to have a new player in the competitive advertiser-supported programming space."
Former Blinkbox director of advertising and sponsorship Scott Deutrom has been appointed managing director of Clubcard TV.
Alongside online, users can watch Clubcard TV on a compatible TV or monitor by connecting up their computer.
But Tesco also plans to launch the service on a range of further devices - including games consoles, tablets, Smart TVs, Blu-ray players and set top boxes - in the future.
The launch of Clubcard TV is part of Tesco's major push into digital entertainment. Earlier in the week, the supermarket announced new music and eBooks services that are forthcoming under the Blinkbox brand.