Can addition cards save money?

I have been told by a financial advisor that carefully selecting and using credit cards can be a method of 'saving money' - by delaying payments by the agreed time limits, then paying without interest charges.
I would like some other inputs and feedback on that idea.

Tesco, Sainsbury, etc. all save money - if used well.
I am talking about the various credit and debit cards which most financial concerns and banks want to sign up their clients to. Are there practical ways to delay spending money, therefore gaining interest which would otherwise be immediately deducted.
The classic theory may be a series of direct debits :- the Barclaycard gets paid by the Lloyds debits, which then get paid by the Tesco credit card etc. All are paid off inside their limits for commencement of charging interest.

It is possible to use a Barclaycard to pay off a debt - at no interest if the Barclaycard is paid within the time limit. Some other cards, e.g. Mastercard, are similar.
The concept works of gaining a bit of 'interest free time' on cards and accounts, providing great attention is made to ensure the outstanding amounts are paid before the interest earning time starts. With enough debt and enough accounts it can earn money - without the essential care and attention it would be a disaster!