Phase 2 of the high speed rail project will see the line run from Crewe to Manchester and the West Midlands to Leeds.
The plan does not include a new station in Sheffield, after proposals to run trains to the Meadowhall shopping centre were shelved.
Ministers have suggested that HS2 should serve the existing Sheffield city centre station, but critics have warned this would mean a new housing estate in nearby Mexborough would face demolition.
The government has claimed that p assengers travelling on the East Coast and West Coast main lines will benefit from more services and extra seats once HS2 is up and running although the current estimate for completion of all of phase 2 is not until 2033.
Phase 1 of the £55.7bn HS2 railway is due to open in December 2026 and will see trains travel at high speed between London and Birmingham before running on from Birmingham on the existing West Coast Main Line.
A second Y-shaped phase will open in two stages. Phase 2a from the West Midlands to Crewe will launch in 2027 and phase 2b, from Crewe to Manchester and from the West Midlands to Leeds, South Yorkshire and the East Midlands, will open in 2033.
Transport secretary Chris Grayling said: "Our railways owe much to the Victorian engineers who pioneered them, but we cannot rest on their legacy when we face overcrowding and capacity problems.
"HS2 is an ambitious and exciting project and the Government is seizing the opportunity it offers to build a transport network fit for the 21st century, one that works for all and makes clear to the world that Britain remains open for business.
"The full HS2 route will be a game-changer for the country that will slash journey times and perhaps most importantly give rail passengers on the existing network thousands of extra seats every day. They represent the greatest upgrade to our railway in living memory.
"But while it will bring significant benefits, I recognise the difficulties faced by communities along the route. They will be treated with fairness, compassion and respect and, as with Phase 1, we intend to introduce further compensation which goes over and above what is required by law."
Former Chancellor George Osborne said he "strongly" welcomed the Government's "renewed commitment" to the second phase of HS2.
He added: "Now let's commit to HS3, the high-speed rail link across the Pennines, so we have a coherent plan to speed journeys across the North.
"We need certainty about the route and effective compensation to help communities directly affected by HS2, including those in my Cheshire constituency.
"Hinkley, Heathrow and HS2/HS3 are the three projects we spent years laying the groundwork for. Time for talking over; now time for delivery."
In June Whitehall spending watchdog the National Audit Office (NAO) warned that the project is under financial strain.
Cost forecasts for phase one exceed available funding by £204m, while phase two contains some elements that are "currently unfunded", the report said.
It went on to warn that the benefit-cost ratio of the scheme could fall from 1.7 to 1.5 if the programme is not delivered well within available funding.
Joe Rukin, campaign manager at Stop HS2, accused ministers of "simply trying to con the public that HS2 is needed for capacity reasons".
He said: "The government have finally come clean in admitting 'freeing up capacity' means for many cities 'losing the trains you already have', as the HS2 business case demands £8.3bn worth of cuts to existing services."
Grayling said a station at the Meadowhall shopping centre on the outskirts of Sheffield was not the government's preferred option.
He told BBC Radio 4's Today programme: "That option has been consulted on before. It clearly remains an option we could follow but the view we have taken is that the plans we have set out today are the ones we should follow."
The cabinet minister said the government was on Tuesday "effectively starting" the process of allowing households hit by the scheme to claim for compensation.
"We want to do the right thing by those people but it is not possible to make the kind of investment we need in the future of our transport system without some impact on people," he added.
Russell Holmes, who lives on the Shimmer estate in Mexborough said residents are still not clear about what is proposed.
He told BBC Breakfast: "That uncertainty obviously is horrible for those having to go through it.
"This estate particularly, people say it's a relatively new estate, its been here three or four years. But it's a really strong community. People get on really well. It's the kind of community where we are proper neighbours.
"It is a proper community that's been built, so for that to be destroyed is devastating."
He said: "But the uncertainty of pretty much knowing it's going to be destroyed but not being able to do anything in the meantime, waiting until it happens, waiting for a decision as to when you can actually put something in place to move or whatever, is horrible as well."
Karen Schofield, who lives nearby, told the programme: "We live just across the bridge on the farm we've been on for 40 odd years.
"Everything has been built up from scratch with our bare hands - a lot of memories, a lot of graft. Not work, graft that's gone into this."
She said: "We're not just talking buying a new house and moving. It's 40 odd years.
"We've never spoken to them, we've had no correspondence from them.
"I would like to say to David Higgins, come to Mexborough, fetch your crew, don't sit on your office with your suit and tie on making all your rules and regulations.
"Come and see what devastation it's going to cause to people."