Russia meanwhile signalled it was ready to contribute up to $20 billion to an EU-led effort to boost the funds available to the International Monetary Fund for rescue programmes which would benefit weaker eurozone states.
Draghi called on EU governments to quickly follow through on their pledge at a summit last week to agree to drastically tougher budget rules.
"The decisions of the European Council summit, together with the (measures) approved recently by the European Parliament, are a breakthrough for clear fiscal rules in our monetary union," Draghi told a congress in the German capital.
"However, the crisis has not ended yet. It is now important not to lose momentum and to swiftly implement all those decisions that have been taken to put the euro area economy back on course," he said.
After non-euro Britain blocked putting budget rules in an EU-wide treaty, the other 26 European Union member states signalled their willingness to draw up a "new fiscal compact" imposing the stricter budget rules.
The pact should be finalised at an EU summit to be held "end January-early February", EU president Herman Van Rompuy said Thursday.
The Czech Republic and Hungary warned however they were against the pact including measures to harmonise tax rates, a perennial issue of dispute among EU nations.
Meanwhile, IMF managing director Christine Lagarde warned the eurozone crisis is still escalating and threatens every economy in the world.
"The issues that we have in front of us now are not just a concern for the eurozone, not just a concern for the European Union, not just a concern for the advanced economies," she told a forum on women in politics at the US State Department.
"There is no economy in the world, whether low-income countries, emerging markets, middle-income countries or super-advanced economies, that will be immune to the crisis that we see not only unfolding but escalating."
With Europe's leaders still struggling after months to come up with a comprehensive fix to end the crisis, Lagarde said the resolution will involve efforts from all countries and regions.
EU plans to boost funds available to the IMF for eurozone rescues got a boost when President Dmitry Medvedev said Russia is "ready to invest."
Medvedev did not name a figure himself but a top aide said earlier Russia could contribute up to $20 billion (15 billion euros) depending on progress by EU nations towards their goal of providing a 200-billion-euro boost to the IMF.
At the summit, EU leaders said they aimed to provide 200 billion euros to the IMF, which would need additional funds to handle a possible bailout of major eurozone economies like Italy and Spain.
However, these plans have been thrown into some doubt as Germany has said it would not provide any extra cash if other non-euro member nations, including Britain and the United States, do not contribute.
Spain's successful raising of 6.0 billion euros ($7.8 billion) in a bond auction, nearly double the planned amount, also boosted sentiment.
"Investors are making clear that they are less dissatisfied with Spain's fiscal position than that of Italy," Rabobank analyst Jane Foley told AFP.
Dealers said sentiment also got a boost from the latest eurozone manufacturing and services survey which showed that a sharp slowdown in the last quarter may be easing, at least in Germany and France.
A series of US indicators on jobs and activity was also positive, especially a sharp drop in new unemployment claims, they said, after heavy losses on Wednesday.
London's FTSE-100 index of top companies closed up 0.63 per cent, while in Paris the CAC-40 rose 0.76 per cent and in Frankfurt the DAX 30 gained 0.98 per cent.
Other markets posted similar gains, with Madrid up 0.84 per cent as Milan put on 1.37 per cent.
In late afternoon trade, the European single currency advanced to $1.3024 from $1.2979 in New York late Wednesday when it hit an 11-month low of $1.2946 after Italy had to pay very high rates to raise fresh funding in the market.
However, hanging over eurozone nations is the threat of an imminent downgrade of their credit ratings, which would likely increase their borrowing costs and make it even more difficult to overcome their debt crisis.
Standard & Poor's is expected to decide soon whether or not to downgrade 15 of the 17 eurozone members after putting them on warning last week.
And rival agency Moody's has said the crisis talks failed to produce "decisive policy measures," saying it would review the credit ratings of all EU states within the next three months.
What the stars mean:
★ Poor ★ ★ Promising ★★★ Good ★★★★ Very good ★★★★★ Exceptional