Chemical Bank, the nation's sixth-largest, cut its prime lending rate below 20 percent today and helped propel the Dow Jones industrial average over the magical 1,000 level.

The Dow average of 30 major industrial stocks climbed 12.03 points to 1004.69, its strongest close since Sept. 27, 1976, when it finished the trading day at 1013.13.

The average barely topped 1,000 on Nov. 20, in the final gasp of euphoria that gripped investors following the Reagan victory in the Nov. 4 presidential election, but it then slid back as rapidly rising interest rates and the apparent start of another recession discouraged investors, at least temporarily.

Bond prices also have rallied sharply in the last three weeks, in large part because of declining interest rates -- first in the open market where banks raise many of the funds they lend, then at the banks themselves.

Although no major banks followed Chemical's lead in cutting the business prime lending rate to 19 1/2 percent, other banks are expected to follow in coming days.

On average, bond prices gained about $2.50 for every $1,000 in face value following a $30 gain Monday. Peter Goldsmith, director of bond research at the giant brokerage firm Merrill Lynch, Pierce, Fenner & Smith, said the bond gain today represented a "consolidation" of earlier gains, with longer-maturing bonds continuing to rise in price.

When interest rates rise, bond prices generally falll and vice versa.

The prime rate, the interest banks charge their top-rated corporate customers for a short-term loan, hit a record 21 1/2 percent a week before Christmas. A few major banks lowered their prime rate to 20 1/2 percent level became widespread only last week.

While the decline in interest rates has helped both the bond and stock markets, it has hurt the dollar. When rates were climbing to their record levels, demand for the dollar swelled as foreign investors rushed to put their funds in U.S. government and corporate securities. Now that interest rates have reached their peak and started to decline, demand for the dollar is beginning to slacken.

The dollar lost ground against most European currencies but rebounded slightly in trading in Tokyo.

In the first three trading days of 1981, the Dow average, the most closely watched of the many stock market indicators, has climbed 40.7 points. The Dow today climbed above 1,000 at noon but lost most of its gain during afternoon trading. A half-hour before the closing 4 p.m. bell, however, a new surge of buying hit the market.

"Some pullback would have been expected," according to Monte Gordon of the Dreyfuss Corp., a major mutual fund company. Not only had there been a rise of 40 points in three days, but many investors also wanted to hold back to see if the 1,000 mark was "for real," he said. "When there was no significant selling pressure, many were emboldened" to buy during the final moments of trading.

The gain in stock prices was broad-based, with 1,055 issues on the New York Stock Exchange closing higher and 632 lower. Volume rose to 67.4 million shares from Monday's 58.7 million on the New York Stock Exchange. It was the ninth-busiest day in NYSE history.

Gordon said the stock market never lost its optimism about Reagan's ability to help the nation's economic posture, despite its concern last month about short-term problems such as high interest rates and money growth. He said that, with the short-term problems diminished, the market is back on its optimistic course. For the long term, he said, the market performance will depend on whether the president-elect is able to translate his campaign promises into an effective fight on inflation and recession.

On the American Stock Exchange, where trading is dominated by smaller technology-company stocks, the index was down 3.68 points to 353.59. However, more Amex stocks rose in price than fell, with gainers outdistancing losers 374 to 292.

On the giant New York exchange, most of the big, so-called blue chip companies rose, including International Business Machines, Du Pont, Procter & Gamble, Eastman Kodak and American Telephone and Telegraph.