Mrs Taylor said she only realised how dangerous the spiders were when pest control experts told her to leave her home.
The family told the Sun newspaper that Sainsbury's initially only offered them a £10 voucher as compensation, but later agreed to pay for the family of four to stay in a hotel while their home was fumigated.
The Brazilian Wandering Spider - also known as the banana spider - is an aggressive and venomous spider found mainly in South America.
It featured in the Book of Guinness World Records in 2010 as the world's most venomous spider - although an effective anti-venom is available and few fatalities occur.
Consi Taylor, her husband and two children were told to leave their home in Hampton in south west London after finding a family of Brazilian Wandering Spiders on some bananas bought from Sainsbury's.
Mrs Taylor told the Sun newspaper that what pest experts later identified as a nest of Brazilian Wandering Spiders she initially thought was a patch of mould.
She told the newspaper: "I thought it was mould but when I had a closer look I saw some funny looking spots. I had a closer look and was horrified to see they were spiders. They werehatching out on the table, scurrying around on my carpet."