The bank has announced 71 Scottish branches and 22 in England will be affected by the changes from July.Read more
BBC Radio 4
Its "fraud refund guarantee" will cover cases where customers are tricked into authorising payments to fraudsters, as well as unauthorised transactions.
Richard Meddings, chair of the bank, told BBC Radio 4's Today Programme that it was the "right thing" to do for customers.
But Victoria Baines, a visiting fellow at Bournemouth University, told the programme that TSB had a tough year after its IT problems and that some of the fraud its customers was facing may have been the result of that situation.
She said the decision by TSB was "admirable" but that rival banks would be looking for unintended consequences.
"There is a possibility this could encourage lax or even fraudulent behaviour on the part of customers," she said.
Customers can take steps to avoid fraud, she said by keeping anti-virus software up to date, not using the same password for banking and social media accounts, and calling back your bank if you get a call.
BBC Radio 5 Live
Wake Up To Money
Criminals stole £1.2bn through frauds and scams from bank customers last year. Today the TSB has promised to refund "innocent victims" of authorised push-payment fraud.
Richard Meddings, acting chief executive of TSB, told Radio 5 Live's Wake Up To Money that the move is "about giving piece 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 is 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!"
Welcome to the first business live page of the week - we'll be with you until around 9.30pm.
News today affecting bank customers - TSB has announced a Fraud Protection Guarantee, ensuring the bank’s 5.2 million customers are protected from all types of third-party fraud losses.
Will the other banks follow? We'll report the reaction to that news.
Meanwhile the numbers of shoppers visiting stores in March was up 1.4% year-on-year, according to the latest British Retail Consortium and Springboard survey.
Do get in touch with your views of any business stories - email us at firstname.lastname@example.org
TSB said that its technology problems cost the bank £330.2m last year including payments for customer compensation and lost income.
The bank said the costs were partially offset by the "recovery" of £153m from Sabis, which is the technology arm of Sabadell, the Spanish company that owns TSB.
TSB's executive chairman Richard Meddings, said: "Last year was TSB’s most challenging year."
He added: "Whilst the migration caused considerable difficulties, we’re now a stronger bank, operating on a more coherent and modern platform, and able to service more customers than ever before."
TSB says that it paid its staff - or "partners" - £1,500 each in December "to recognise the exceptional team effort across the business to put things right for customers" following the bank's difficulties in implementing a new IT system.
It said that its executive would not receive a bonus and its new chief executive, Debbie Crosbie, will join the company in the spring.
TSB's former boss Paul Pester resigned last September following the IT meltdown at the bank.
TSB has just announced a pre-tax loss of £105.4m in 2018, which it blamed on its IT meltdown last year. That compares to a £162.7 pre-tax profit in 2017.
Difficulties with introducing a new computer system last year meant that large numbers of TSB customers were unable to access their own money.