Members of 13 unions representing hospital cleaners, nurses, security guards, physiotherapists, emergency call handlers, paramedics, midwives, radiographers and other NHS staff across England voted to accept the deal.
The GMB is the only union involved in the NHS which has rejected the offer.
Unison's head of health, Sara Gorton, said: "The agreement won't solve all the NHS's problems overnight, but it will go a long way towards easing the financial strain suffered by health staff and their families over many years.
"The lifting of the damaging 1% cap on pay will come as a huge relief for all the employers who've struggled for so long to attract new recruits and hold on to experienced staff.
"But this three-year pay deal must not be a one-off.
"Health workers will want to know that ministers are committed to decent wage rises across the NHS for the long term, and that this isn't just a quick fix.
"Most importantly, the extra funding means the pay rise won't be at the expense of services or patient care.
"Now the Government has begun to put right the damage inflicted by its mean-spirited pay policies, staff will be hoping ministers announce an injection of cash for NHS services in time for its 70th birthday next month."
Unions held a series of consultations with NHS staff on the offer and announced that they voted overwhelmingly to accept the deal.
NHS staff should now get the money in their July pay packets, backdated to April.
The agreement, reached after months of negotiation between unions, employers and ministers, was made possible with an extra £4.2 billion of government funding.
Unions said the decision to accept the deal means a significant wage boost for the lowest paid workers in the NHS.
Hospital caterers, porters, administrators and other staff on the lowest grades will receive a wage rise of more than £2,000 this year - an 11%-13% increase.
Every NHS worker in England will now be paid at least £8.93 an hour, or £17,460 if they work full-time.
Josie Irwin, of the Royal College of Nursing, said: "Today's deal gives a much-needed pay rise to over a million people and, at a time when there are 40,000 unfilled nurse jobs in England alone, it should help to make the profession more attractive to current and future nurses alike.
"By standing together, the NHS unions were able to reject all unpalatable demands to cut annual leave or unsocial hours payments and get the best possible deal from a Government still committed to austerity.
"We have taken a significant step on the journey towards fairer pay for NHS staff but there is much more to achieve, not least for the staff who deliver NHS services outside direct employment.
"The Government would be mistaken if it thought today's deal was the end, rather than the beginning, of that journey."
It is expected that additional funding will now be made available for health budgets in Wales, Scotland and Northern Ireland, allowing pay negotiations to take place for health workers there.
GMB officials will meet next week to decide their next move after members rejected the deal by almost 9-1.
Claire Sullivan, of the Chartered Society of Physiotherapy, said: "This agreement gives NHS staff in England long overdue recognition of the enormous contribution they have made in recent years under huge pressure."
Gill Walton, general secretary of the Royal College of Midwives, whose members voted by more than 8-1 in favour, said: "Midwives, maternity support workers and other NHS staff can start to recoup the losses of the last eight years.
"It is also important that, apart from this significant pay increase, we successfully fended off employer attempts to reduce unsocial hours payments and annual leave.
"This is the best deal in the public sector and the only one which is additionally funded, meaning it will not come out of existing NHS money."
The unions whose members voted to accept the agreement are the British Association of Occupational Therapists, the British Dietetic Association, the British Orthoptic Society, the Chartered Society of Physiotherapy, the Federation of Clinical Scientists, Managers in Partnership, Prison Officers Association, the Royal College of Midwives, the Royal College of Nursing, the Society of Chiropodists and Podiatrists, the Society of Radiographers, Unison and Unite.
Health and Social Care Secretary Jeremy Hunt said: "This is an incredibly well-deserved pay rise for staff who have never worked harder.
"Salaries will increase from between 6.5% and 29%, with some of the biggest increases for the lowest paid.
"I hope this will also go some way to helping us recruit and retain more brilliant staff in our NHS."