A burst of selling at the closing bell drove the Dow lower after it hovered around the milestone for most of the afternoon. The average finished the day about 19 points shy of the mark.
The Dow broke through 13,000 several times last week but hasn't closed above that level since May 19, 2008, four months before the fall of Lehman Brothers investment bank and the worst of the financial crisis.
For the day, the Dow lost 1.44 points and closed at 12,981.51. The Standard & Poor's 500 index rose 1.85 points to 1,367.59, a 3½-year high. The Nasdaq composite index rose 2.41 points to close at 2,966.16.
The Dow fell 100 points at the open Monday, then climbed back above 13,000 after a report that the number of Americans who signed contracts to buy homes rose in January to the highest in almost two years.
The National Association of Realtors said its index of sales agreements rose 2 percent last month to a reading of 97, the highest since April 2010. A reading of 100 is considered healthy.
Financial stocks led the industries gaining ground Monday. They rose 0.9 percent as a group. All nine of the other industry groups in the S&P 500 finished with small gains or narrow losses.
Scott Wren, senior equity strategist for Wells Fargo Advisors in St. Louis, said investors have gotten ahead of themselves since October. The S&P 500 is up 8.8 percent this year alone.
He said he thinks U.S. economic growth is likely to be a mild 2 percent this year, there are fewer people working now than there were at the end of 2007, and Europe may be in a recession.
"I don't see any reason for the market to be on some incredible run," he said.
The price of oil fell below $109 a barrel as investors booked profits after a 14 percent gain this month driven by signs of an improving U.S. economy and fears of an Iranian supply cut. Benchmark crude fell by $1.21 to end the day at $108.56 per barrel in New York.
Brent crude, which is used to price oil that's imported by U.S. refineries, lost $1.30 to finish at $124.17 per barrel in London.
Government figures show that growth in demand for crude oil has slowed in the U.S. from a year earlier, although some oil traders are betting a strengthening economy will eventually boost consumption.
"Four dollar gas, that's not going to really shut down the economy," Wren said. "$5, $6 gas, that's a different story."
The average U.S. price of a gallon of gasoline has jumped 18 cents over the past two weeks, with a gallon of regular at $3.70.
Overseas markets fell slightly because of worries about high oil prices. Stocks fell 0.3 percent in Britain, 0.2 percent in Germany and 0.7 percent in France.
The European debt crisis is also still a lingering concern. Finance ministers from the world's 20 leading economies said that they would not add money to the International Monetary Fund until the European Union puts up more money to stave off its debt crisis.
Earlier in Asia, Japan's Nikkei 225 index ended down 0.1 percent at 9,633.9, giving up gains posted earlier in the day. Hong Kong's Hang Seng fell 0.8 percent to 21,217.86 and South Korea's Kospi lost 1.4 percent to 1,991.16.
Other commodities rose, including a 1.2 percent rise in wheat prices in Chicago. Soybeans topped $13 a bushel for the first time in five months. Global soybean supplies are tight, and hot, dry weather has damaged crops in Brazil and Argentina.
Among U.S. stocks making moves:
— Cooper Tire & Rubber Company Co. rose 13.9 percent after it said net income more than quadrupled in the fourth quarter, mostly because of a large tax-related gain.
— Chipmaker Micron Technology Inc. rose 7.7 percent after a Japanese rival company filed for bankruptcy protection.
— Warren Buffett's Berkshire Hathaway Inc. rose 0.3 percent after the investor, known as the Oracle of Omaha, said the company has a successor in mind, plus two backups. Berkshire has not named those executives.
— Netflix stock fell 2.2 percent after a downgrade by an analyst at Raymond James.
— Shares of Sprint Nextel Corp. rose 3.2 percent after reports that its board rejected a purchase of MetroPCS Communications. MetroPCS fell 1.5 percent.
— Lowe's, the home improvement chain, rose 0.7 percent after reporting a 13 percent jump in fourth-quarter profits.