Across England, 490,000 (77%) pupils met the required standard, up 120,000 on 2012, when the tests began.
The National Union of Teachers said the jury was "still out" on the "obsession" with one teaching method.
But the schools minister pledged to challenge councils with poor results to come up with swift plans to improve.
Nick Gibb said the results showed the focus on phonics was ensuring children were becoming "confident, inquisitive and fluent readers".
The top performing local authority was Richmond-upon-Thames, where 86.5% of six-year-olds made the grade, followed by the London boroughs of Kensington and Chelsea, Lewisham, Wandsworth, Newham, Greenwich, Harrow, Bromley and Bexley - with Darlington, on 82.3%, the only non-London borough in the top 10.
By contrast only 69.5% of six-year-olds met the standard in Nottingham, the worst performing council - followed by Peterborough, Wakefield, Doncaster, Derby, Liverpool, Bury, Oldham, Middlesbrough and Tameside.
Mr Gibb said: "Today's results demonstrate the effectiveness of the government's continued focus on phonics as the primary way of helping young people to read.
"For years, children were being denied the joy of becoming fluent readers because of a reliance on teaching methods that failed too many children.
"We will continue to challenge those local authorities whose phonics results are below the levels achieved elsewhere.
"The evidence is clear that the systematic teaching of phonics is the most effective way to help children master the basics of reading so they can go on to become confident, inquisitive and fluent readers."
But NUT general secretary Christine Blower said the point was not whether test scores were rising.
"It's whether the government's decision to focus the teaching of reading on one method, synthetic phonics, is producing an improvement in children's literacy.
"Here, the jury is still out.
"The [Department for Education's] own study, published earlier this year, found no conclusive evidence that any improvements are attributable to the check."
This "very narrow approach to learning to read" risked sidelining reading for pleasure, Ms Blower added.
Nine of the 10 local authorities in which most six-year-olds passed the government's phonics reading test this year are in London, figures show.