The Labour leader trumped Mr Johnson in the popularity stakes by by two percentage points, according to the Opinium poll.
It found that 37 per cent of voters thought Sir Keir would be better at leading the country, compared to 35 per cent for the current PM.
Sir Keir has been enjoying a higher approval rating than his Tory rival for the past six weeks, however, it is the first time he has outperformed Mr Johnson when asked who would make the better Prime Minister, the pollsters said.
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 per cent said they would vote Tory at the next election, compared to 39 per cent for Labour.
Both parties are down one percentage point since last week.
The poll also found that there is strong public disapproval of the Government’s latest handling of the coronavirus crisis, with the majority fearing restrictions are being lifted too quickly.
The survey was carried out in the first week of June, prior to the announcement of a major relaxation of social distancing measures from July 4.
According to Opinium, 54 per cent of Brits believe the country is exiting lockdown too quickly – up eight points from last week.
The weighted results are based on an online survey of 2,002 UK adults aged 18 or over, conducted between June 4-5.
Adam Drummond, head of political polling at Opinium, said: “Boris Johnson’s approval rating remains narrowly negative, while Keir Starmer’s remains substantially positive (moving from a net 22 last week to 27 this week).
“Politically, the ‘new normal’ is that for the first time in over a decade Labour has a leader who outperforms their party while the Conservatives have a leader who under performs theirs, and the crossover in preferred prime minister is a reflection of that.”