The former leader of the party, who resigned in 2015 after a heavy election defeat to David Cameron, was asked about his successor's potential during an appearance on the BBC’s Andrew Marr Show.
The Oxford graduate controversially won the 2010 leadership contest due to trade union backing, despite MPs and party members preferring his brother and rival David Miliband.
He was heavily criticised and returned to the backbenches after the Tories won a surprise 12-seat majority at the 2015 poll.
Miliband returned to the shadow cabinet in April this year after Sir Keir made him shadow business secretary.
Host Andrew Marr asked Miliband on Sunday: "One slightly mean question to end with.
“You've seen Keir Starmer now in operation for a few months. Is he going to be a better leader of the Labour Party than you?"
Miliband replied: "Definitely. I think you've seen that already. Look, I certainly never had his approval ratings.
"I think he's made a great start. I think he's shown not just competence but the kind of seriousness that this crisis demands.
"I think the more people see of him the more they'll see the integrity, the principle and decency I know really well."
It comes as an Opinium poll for The Observer found that 37% of voters thought Sir Keir would be better at leading the country than current Downing Street incumbent Johnson.
A total of 35% thought Tory leader Johnson was the better option to have in charge.
The pollsters said that while Sir Keir had been enjoying a higher approval rating than the prime minister for the past six weeks, it is the first time the former director of public prosecutions has polled higher when asked about who would make the better PM.
Sir Keir replaced Jeremy Corbyn as Labour leader in April following a disastrous election defeat for the party in December.
The Conservatives, according to the results published on Saturday, still hold a four-point lead in terms of voting intentions, despite Sir Keir's public popularity.
Out of those polled, 43% said they would vote Tory at the next election, compared to 39% choosing Labour.