The arrangement is expected to raise a total of at least £5m for the two charities, which take over from the children's cancer charity Clic Sargent.
The sister charities submitted a joint bid, with the winner chosen by a Tesco committee.
The deal means the charities will receive income from fundraising by Tesco staff and from special events, such as collections in Tesco stores over the bank holiday weekend at the end of May and other activities linked to major events, including the royal wedding in April.
Money raised in Tesco stores in Scotland will go to Alzheimer Scotland; money raised in stores in England, Wales and Northern Ireland will go to the Alzheimer's Society.
The charities have today unveiled the results of research revealing differences in dementia care across the UK.
David Reid, chair of Tesco, said: "There are 750,000 people living with dementia. This means that almost half of our staff and customers have a loved one or friend with dementia, so we are proud to have the opportunity to work with the Alzheimer's Society and Alzheimer Scotland to make a real and lasting difference to their lives, from diagnosis through to treatment."
Jeremy Hughes, chief executive of the Alzheimer's Society, said: "The money raised through the Tesco partnership will help us ensure people with dementia don't face it alone."