He succeeds Andrew Gillespie, who held the role on an interim basis, and will report to chief marketing officer Nigel Gilbert. Markey is expected to start in June.
Markey joined Aviva in March 2016 after two years as chief marketing officer at the Post Office. He previously worked for rival insurers RSA Group, where he spent eight years, and More Th>n.
He told Campaign that the chance to work with Gilbert, who he called an "inspiration", had been a key factor in the decision to join TSB.
Markey added: "It is a really exciting time to join TSB as it continues to challenge the UK banking sector and I can’t wait to get stuck in.
"I’ve been impressed with how the bank has established its personality within the sector, and in my new role, I’m looking forward to helping evolve the brand as TSB grows."
Markey, a member of Campaign's Power 100 Hall of Fame, was an early champion of the use of data, insight and intelligence.
At Aviva he also co-chaired Aviva Pride, the insurer’s lesbian, gay, bisexual, and transgender employee network.
Gilbert said: "It’s nearly four years since we launched TSB as Britain’s challenger bank, and in that time, we’ve become the most recommended high street bank in Britain, we’re growing, and more people than ever are considering TSB as a bank worth switching to.
"As we continue to drive this momentum, I am really thrilled to bring Pete on board. With all of his rich experience, Pete will play a major role in the next phase of our marketing and brand strategy."
Aviva confirmed that Markey left the insurer at the end of April, and that as part of structural changes, he would not be directly replaced.
A spokesman said: "Pete has made a valuable contribution to our marketing and brand performance over the past year and we would like to thank him for all of his work since joining the business and wish him well in his future endeavours."