The north tower will be the first to reach 202m (673ft), with the others following over the next two months.
The Scottish Government’s Transport Scotland agency, which is in charge of the £1.4 billion project, said this is expected to happen in the next ten days.
However, the centre tower will be even taller, at 210m (700ft), with the south tower also 202m.
The crossing will then dwarf the two towers of the adjacent Forth Road Bridge, which are 50m (166ft) lower.
The previous tallest towers were those on the Humber Bridge, at 162m.
Sections of the new bridge’s deck weighing 750 tonnes each - as heavy as a small Royal Navy ship - are also being lifted into place on the towers.
Infrastructure secretary Keith Brown, who watched progress from North Queensferry today, described the operation as “enthralling”.
He said: “On a project featuring many spectacular sights, the lifting of 750-tonne deck segments from barges on the Forth up 200 feet to the deck level is really something extra special.
“The precision and complex engineering required to achieve this is equally enthralling.
“Local people and commuters on the Forth Road Bridge will be treated to this spectacle over the next year or so and it will only get more spectacular as the road deck fans out from the towers, supported by newly-fitted cables as they go.”
The first of 110 deck units were installed last month and are expected to take a year to complete.
The sixth section was lifted into position today, at the north tower.
There are now four sections in place there and two at the south tower, with deck lifting expected to start at the centre tower this month.
The bridge is due to open to traffic by the end of next year, leaving the Forth Road Bridge reserved for bus and taxis.
It was ordered following doubts about the long-term strength of the 51-year-old bridge after the discovery of corrosion of its main support cable.