Mr Saha's monthly salary as a school teacher in the eastern state of West Bengal is 35,000 rupees ($700).
But his account balance of $9.8bn is closer to India's annual education budget of $11.5bn.
"On Sunday evening, when I was checking my savings account balance on the internet, I was expecting an amount of a little more than 10,000 rupees ($200)," Mr Saha, who lives in the town of Balurghat in South Dinajpur district, told the BBC.
The billions he found there instead have left him in shock.
"I called up a friend in the bank and joked, maybe money is overflowing in your bank, that's why your system has remitted so much money into my account," he said.
The government-run bank - motto "Safe Banking with SBI" - has not explained what happened.
Subhashish Karmakar, chief manager of the bank's branch in Balurghat, declined to speak about the source of the funds or how the money got into Mr Saha's account.
"I have been specifically asked not to comment on this issue," he told the BBC.
The bank's regional headquarters in Calcutta and national headquarters in Mumbai have been alerted and officials are trying to figure out what went wrong, sources in the bank say.
Mr Saha says he has managed to withdraw the $200 from his account that belonged to him.
"Even though I got my money back, the account still holds the billions of dollars as uncleared amount. I don't know how long I'll have to keep that astronomical figure in my account."