Mario Draghi’s big bond-buying plan won’t start until March, but it already is a big hit with investors.

Traders and investors took some time to warm up to the news Thursday that the European Central Bank will buy at least €1 trillion ($1.16 trillion) of the region’s bonds, including first-time purchases of government debt, from March 2015 to September 2016. But many eventually concluded the stimulus program was big enough to help prop up financial markets on both sides of the Atlantic.

Stocks in Europe and the U.S. swung around as Mr. Draghi, the ECB president, took to the stage at a news conference in Frankfurt to lay out the €60 billion-a-month bond-buying program and field questions. Later, after investors had time to digest some of the details, stocks rallied broadly, and U.S. markets continued to rise after the Stoxx Europe 600 ended at a fresh seven-year high. The Dow Jones Industrial Average jumped 259.70 points, or 1.5%, to 17813.98, and as of Thursday’s close was just 1.3% off its all-time high hit last month.

The euro teetered on the brink of previous 11-year lows and then plunged through them, notching its biggest one-day loss against the dollar since November 2011. In late New York trading, one euro bought $1.1368, off 2.1% from $1.1611 Wednesday.

Yields on many European government bonds slid to new record lows as investors scooped them up ahead of the program’s start. One of its aims is to lower bond yields, which spurs investors on the hunt for returns into riskier assets such as stocks. When bond yields fall, prices rise.