Over 2,000 people were interviewed for the firm's Family Finance Report.
The average amount saved was £96 per month and the average total savings pot was £3,281. The monthly rates dropped as low as £19 in November 2011.
The number of families failing to save at all reached an all-time low of 23 per cent, down from 33 per cent in January 2011.
Some 41 per cent have an Isa while 21 per cent of families have premium bonds but only six per cent have fixed-term bonds.
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Some 17 per cent were investing in stocks and shares, up from 11 per cent in August 2012.
For parents with one child, investments in stocks and shares increased from nine per cent to 19 per cent. Aviva said this suggests people are thinking more about their finances once they have dependents.
Over a third have a workplace pension and 18 per cent had a private pension.
Louise Colley, protection distribution director at Aviva, said: "Given the atmosphere of austerity that has lingered since the Family Finances Report began, the renewed commitment to saving bodes well for the future.
"Despite the pressure on the purse strings, more families are take steps to provide themselves with a financial safety net. By considering a range of savings and investment products, they can find a favourable return and make sure their money works as hard as it can for them."

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