The austerity and accompanying economic reforms have hit living standards hard and contributed to huge increases in unemployment.
The country is now seeing some rays of hope, though the recovery is expected to be slow.
Staikouras said four years of hardship and sacrifices "are beginning to produce a result, creating the first indications of Greece exiting the crisis."
Such bold talk would have been unimaginable just over a year ago, when many analysts were expecting Greece to ditch the euro amid political uncertainty and foot-dragging on the reform agenda. Since then, its creditors have eased loan conditions and promised further assistance once Athens achieves a primary budget surplus that is, makes more than it spends before counting the cost of debt servicing.
Finance Minister Yannis Stournaras, speaking on a late-night television news program, stressed that Greece was beating the targets it had set for reforming its economy.
"It's the second consecutive year that we have results that are better than the targets," Stournaras said.
Fears remain that the bailout program may fall short of Greece's financing needs by about 10 billion euros ($13.5 billion) in 2014-15, although Athens argues that the gap can be filled.
The draft budget Staikouras presented foresees a slight drop in unemployment, from 27 percent in 2013 to 26 percent next year. That is still not far from the 28 percent hit in June, which is the highest rate among the 17 countries that use the euro currency.
Staikouras warned the government cannot afford to relax its five-year austerity drive, with further reforms and privatizations needed. Athens continues to face a crushing debt load, expected to reach 174.5 percent of its annual gross domestic output next year, down from 175.5 percent in 2013.
Staikouras said long-term debt sustainability "remains an open issue," as Greece has committed to reduce its debt-to-GDP ratio to below 110 percent in 2022. He said a mixture of primary surpluses, further reforms and privatizations combined with additional aid from creditors would help ease the load.
The deputy minister said the conservative-led government has been able to reduce the budget deficit to 2.4 percent of output this year, from 6 percent in 2012 and more than 15 percent three years ago. According to the draft budget, the deficit will remain at 2.4 percent in 2014.
The primary surplus is forecast to rise from a modest 340 million euros this year to 2.8 billion in 2014.
"These positive developments must add impetus to our efforts and not allow any relaxation, because Greece's chronic structural and fiscal problems ... have not been fully addressed," Staikouras said at a press conference.
Staikouras confirmed earlier forecasts that Greece hopes to once again be able to borrow in international bond markets late next year. "It is our intention, and we are taking initiatives to ensure that access to markets becomes feasible in the second half of 2014," he said.
Stournaras, speaking later in the night, would not be pinned down on specific timing, however.
Asked for how much longer Greece's economy would be under international supervision, the minister said, "The troika will leave when we can stand on our own two feet and go back to the markets." Asked when that would be, he said simply, "Soon."
The final version of the budget will be submitted to Parliament following extensive negotiations with Greece's bailout creditors, and is expected to be voted on in December.