Stock prices rallied sharply in late, heavy trading today, and the Dow Jones industrial average climbed above the 1,000 barrier for the third time since Ronald Reagan was elected president and the 17th time in stock market history.

The Dow average -- the most widely known barometer of stock prices -- finished the day at 1,002.79, up 17.02 points. Standard & Poor's 500-stock index climbed 1.57 points to 134.68.

Analysts said the strong rally was due to a variety of factors, including the expectation of further declines in interest rates, an orderly slowdown in the economy that should help reduce inflationary pressures and the apparent ability of the Federal Reserve Board to contain the growth of the money supply despite an unexpected surge in money growth last week.

But today's late rally was touched off by big institutional investors such as pension funds and insurance companies, according to Jerry Hinkle, manager of trading at the brokerage firm Sanford C. Bernstein & Co. After last Thursday's sharp 22.15-point climb, most institutions held back from buying to see whether stock prices would fall back sharply, Hinkle said.

"But the market gave little ground, and Friday's pullbck was orderly," according to Newton Zinder of the brokerage firm E. F. Hutton & Co. On Friday the Dow average fell 4.05 points in heavy trading -- although most of the buying was concentrated in the morning. Today stock prices moved little in light trading until well after lunch.

"Then new buyers were attracted in, and [the rally] began to feed on itself. Momentum is an important thing in the market," Zinder said.

But investors also were heartened by a continuing decline in short-term interest rates.

On the New York Stock Exchange, 1,085 issues closed higher in price, while 499 declined. Composite volume for all NYSE-listed stocks in U.S. trading today was 57.2 million, down from the 76.8 million shares that changed hands Friday.

The New York Stock Exchange's index was up 0.87 point to 77.19.

On the American Stock Exchange, where the index closed up 3.05 points to 347.47, there were 392 stocks that closed higher while 231 declined. Volume was 5.8 million shares compared to 6.8 million on Friday. The National Association of Securities Dealers' NASDAQ index of over the counter issues gained 1.27 points to 203.44.

St. Joe Minerals, which is fighting a takeover attempt by Seagram's, fell 2 1/8 to 45 3/8, while Seagram's fell 1/8 to 54 3/4. Standard Oil of Ohio, which agreed to buy Kennecott Corp. for $62 a share, fell 1/2 to 51 1/2, and Kennecott declined 1 3/4 to 52 1/4. In the other major takeover attempt, Amax, which rebuffed a bid by Standard Oil of California, closed down 3 1/4 at 55, while Socal closed up 5/8 at 42 1/4.

Although the takeover candidates and their suitors did not fare well today, most other stocks did. Bethlehem Steel closed up 1 3/8 to 30 1/8, and U.S. Steel finished the day at 32 3/4, up 1 3/4.