In Spain, the Ibex share index closed up 6%, while in Italy, the main share index closed up 5.6%.
The comments also triggered a fall in bond yields. Spain’s 10-year yield, which had hit a record high of 7.6% earlier, fell back to 6.8%.
Bond yields are an indication of the interest rate a country would have to pay to borrow money. A rate above 7% is generally seen as unsustainable in the long run.
Earlier in the week, share prices had dived and Spanish bond yields jumped sharply on fears that the debt problems being faced by several of the country’s regional governments would push Spain towards a full bailout.
Mr Draghi’s comments also boosted the euro, which had been languishing at two-year lows against the dollar earlier this week. It rose 1% to $1.2284.
In other stock markets, the UK’s FTSE 100 ended up 1.4%, France’s Cac 40 jumped 4.1% and Germany’s Dax climbed 2.8%.
On Wall Street, US stocks rose in early trade. The Dow Jones was up 1.5%.
Speaking at a conference in London marking the start of the Olympics, Mr Draghi said: “Within our mandate, the ECB is ready to do whatever it takes to preserve the euro. And believe me, it will be enough.”
Many, investors took that to be a strong hint that the ECB might re-start its Securities Markets Programme (SMP).
The scheme – introduced in 2010 – allows the ECB to buy large quantities of government bonds from banks and other financial institutions on the open market.
That, in turn, allows indebted eurozone governments to borrow money at rates much lower than those offered in the commercial bond markets.
The last SMP purchase took place at the end of January. Since then, Mr Draghi has, repeatedly reiterated his resistance to large-scale bond purchases. The last time was at the ECB’s 5 July news conference.
The ECB holds its next interest rate meeting and news conference next week.
An auction of Italian two-year bonds on Thursday saw the interest rate investors demanded to lend to the Italian government rise to an eight-month high of 4.86%.
However, demand for the bonds was strong, and the yield on the benchmark 10-year bond fell back in line with Spain’s – another positive sign.
The French finance minister, Pierre Moscovici, said Mr Draghi’s remarks on government bond yields were “very positive”.
Elsewhere in Europe, the President of the European Commission, Jose Manuel Barroso visited Greece for the first time in three years for talks.
“The prime minister has assured me that the coalition government will respect commitments… and will speed up the key structural reforms that are needed, including privatisation and of course reforms in the public administration,” said Mr Barroso late on Thursday in Athens.
Greece, which is the worst-affected country in the eurozone, is also playing host to international monitors from the European Union, the International Monetary Fund and the European Central Bank, who are there to check how it is getting on with its debt-cutting programme – a condition of its bailout loans.
The three, known as the troika, had a two-hour meeting with the Greek finance minister Yannis Stournaras.
A senior official said that the government had identified proposed spending cuts for the next two years, which he said were likely to be presented to the inspectors later on Thursday.
The euro crisis has spread its negative effects even further this week, with the credit ratings agency Moody’s downgrading the outlook for Germany’s AAA credit rating to negative, because of the prospect of it having to provide more help to prop up the single currency.