TSB has become the first UK bank to pledge to refund customers who fall victim to any type of fraud.
The "fraud refund guarantee" will cover cases where customers are tricked into authorising payments to fraudsters, as well as unauthorised transactions.
The move comes as the bank tries to rebuild its image after an IT meltdown last April left 1.9 million customers unable to access their own money.
Banks have been under pressure to help tackle the rise in sophisticated fraud.
Richard Meddings, acting chief executive of TSB, told Radio 5 live's Wake Up To Money that the move was "about giving peace of mind to our customers and doing the right thing".
He said: "It's a major societal blight. Innocent customers are being tricked."
He added that the bank was investing in education for customers and staff about fraud, but also had a message for crooks: "If you come for one of my customers, we will hunt you down."
Currently, victims who are tricked into transferring money directly from their account to a fraudster are less likely to be reimbursed, because they approved the payments.
Some £354m was lost last year through this type of scam, known as a "push" or "authorised" payment fraud, according to banking trade body UK Finance.
Financial firms returned just £83m of this to customers.
Examples of authorised payment fraud include fraudsters posing as builders, solicitors, or other contractors who have carried out work for the victim. They submit a fake invoice containing the fraudster's bank details and it is often not easy to spot that they are not the legitimate payee.
"The vast majority of fraud claims across UK banking are from innocent victims of fraud who have been targeted by criminals and organised gangs.
"However, all too often these customers must fight to be refunded and are not treated as victims of crime," said TSB executive chairman Richard Meddings.
TSB said its guarantee - which applies to losses from 14 April - marked a "step change" in the industry, where currently customers were only refunded for fraud losses in limited circumstances.
Under the guarantee, customers will need to contact the bank to report fraud and it will still investigate the fraud claim, including what happened and how, so it can inform customers and ensure they are protected from future fraud.
The bank, which has 5.2 million customers, warned it would not reimburse customers who tried to abuse the guarantee by committing fraud on their own account or by repeatedly ignoring safety advice.
Last month, banks and building societies agreed to do more to protect customers, introducing a new voluntary code which comes into effect on 28 May.
But consumer watchdog Which? said banks needed to do more.
"Other High Street banks are leaving their customers unprotected. All banks must now follow TSB's lead and ensure that their own customers are not left paying for the cost of this crime," said Jenny Ross, Which? money editor.