The announcement came as Downing Street unveiled a 1.7% hike for prison officers and improvements totalling 2% in police pay for 2017/18.
The end of the ceiling on public sector rises came after massive pressure from unions and Labour, and was made public shortly before Jeremy Corbyn rose to address a TUC conference in Brighton dominated by calls for an end to restraint.
Unions made clear in their initial responses that the Government move fell well short of their aspirations.
TUC general secretary Frances O'Grady branded the increases for police and prison officers "pathetic", on a day when the latest inflation figures showed prices rising by 2.9% annually.
The POA prison officers' union said it was seeking industrial action over an offer which it said effectively amounted to a pay cut and would leave a majority of staff with rises of just 1.3%.
Meanwhile, police chiefs warned that the pay award would put financial pressure on forces' already-stretched budgets and could impact on their ability to deliver services and avoid job cuts.
Police will receive a 1% one-off "non-consolidated" bonus on top of their basic pay rise of 1% for 2017/18.
Their settlement and the 1.7% average rise for prison officers were agreed by Cabinet in line with the recommendations of the last two independent public sector pay review bodies to report this year.
The 2017/18 settlements will be met out of existing departmental budgets and will be implemented immediately.
The Government's pay restraint policy has seen a two-year freeze after the Conservative-led coalition came to power in 2010, followed by a 1% annual limit from 2013.
Asked whether the readiness to show "flexibility" meant that the cap was now ending, the spokesman said: "The answer is yes."
"There will still be a need for pay discipline over coming years to ensure the affordability of public services and the sustainability of public sector employment," he said.
"However, the Government recognises that in some parts of the public sector, particularly in areas of skill shortage, more flexibility may be required to deliver world-class public services, including in return for improvements to public sector productivity."
Chief Secretary to the Treasury Liz Truss said review bodies and departments were being given greater leeway to use pay to address "pinch points" within public sector staffing.
"We are making sure that our policy is targeted to where there are specific issues, where we need to make sure we recruit more talent into the public sector, but also where we do need to make sure that we are holding on to those really valued people," Ms Truss told BBC Radio 4's World at One.
"What we are making sure is that we look at it on a workforce-by-workforce basis because there are very different issues for teachers than for nurses and for police officers."
Mrs May's spokesman said ministers meeting in Cabinet at Downing Street on Tuesday "agreed that public sector workers are among the most talented and hard-working people in our society. They, like everyone else, deserve to have fulfilling jobs that are properly rewarded".
He added: "Government will continue to ensure that the overall package for public sector workers recognises the vital contribution they make and ensures we can deliver world-class public services, while also being affordable within the public finances and fair to taxpayers as a whole."
Ms O'Grady described the below-inflation rises for police and prison officers as "derisory".
"Public sector workers have suffered seven long years of real pay cuts, and are thousands of pounds worse off," she said.
"Today's announcement means bills will continue to rise faster than their wages. If ministers think a derisory rise like this will deal with the staffing crisis in our public services, they are sorely mistaken."
The Police Federation of England and Wales, which had asked for a 2.8% uplift to basic pay, warned the announcement will leave many officers "angry and deflated".
"Police officers do not join the service to make huge amounts of money, they do it out of a sense of duty and this year in particular have been tested to the max," said chairman Steve White.
"However, they expect to be paid suitably for the immensely demanding role they perform and this simply is not the case."
National Police Chiefs' Council spokesman Francis Habgood warned: "Police chiefs have budgeted in line with the public sector pay cap until 2020 so this change puts financial pressure on already-stretched budgets.
"Without better real-terms funding protection from Government, an award above 1% will inevitably impact on our ability to deliver policing services and maintain staffing levels."
Liberal Democrat leader Sir Vince Cable said: "It is good to see the Government finally recognise that the public sector pay cap is no longer sustainable.
"The cap must now be lifted across the board so all public sector workers are given the pay rise they deserve."
Mr Corbyn denounced the Government's announcement as an attempt to play "divide and rule" with public sector workers, and promised that Labour would ditch the cap across the board.
The Labour leader told the TUC conference in Brighton: "This Government's position seems to change by the hour. At the weekend we were led to believe the pay cap was a thing of the past. Yesterday, the Prime Minister's spokesman said it would continue as planned. Today, as inflation rises to nigh on 3%, they try to divide people on the cheap.
"The POA is right. A pay cut is a pay cut. We must be united in breaking the pay cap for all workers.
"So let me be absolutely clear today. The Labour Party totally rejects the Tories' attempt to divide and rule, to play one sector off against another. A Labour Government will end the public sector pay cap and give all workers the pay rise they deserve and so desperately need. That is our policy."