The nation's economy grew in the final quarter of last year at a full 5 percent annual rate after adjusting for inflation, the Commerce Department said yesterday in a preliminary report. This was faster than the 2.4 percent growth recorded in the previous three months and was higher than expected by officials even a few weeks ago.

At the same time, the department said that the underlying rate of inflation in the economy accelerated somewhat from the third-quarter rate. The so-called GNP deflator, which reflects the changing prices of what actually is produced and bought, went up at an annual rate of 11.2 percent in the fourth quarter compared with a rise of 9.2 percent in the previous three months.

Inflation on this index accelerated from 8.1 percent in 1979 to 9 percent last year.

The gross national produce, a measure of the total value of the goods and services produced in the economy, actually fell by 0.1 percent during 1980 as a whole from the level in 1979. But the drop was much less than first expected. The economy recovered so quickly from recession in the second half of the year that by the end of last year output was almost back up to its level of a year earlier.

Many economists expect the upturn to trail off this quarter, with perhaps a drop in growth recorded. This is because of the swift climb in interest rates last month and the dampening effect which this has on the economy. However, so far there have been few signs of a slowdown, except in the interest-rate-sensitive auto market.

Yesterday's figures showed that personal disposable income, after taking account for inflation and taxes, rose by a 3.2 percent annual rate in the fourth quarter. The savings rate, savings as a proportion of income, dropped slightly from 6.1 percent in the third quarter to 5.6 percent in the final quarter as personal spending rose by $62.2 billion.

The Carter administration last week predicted a stagnating economy in the first half of this year, followed by a 3 1/2 percent annual rate of growth in the last six months of the year. The new administration is committed to an early tax cut; if this is enacted without compensating cuts in federal spending, then there may be faster growth than forecast by the last administration.

Final sales, adjusted for inflation, rose by a 3.7 percent annual rate in the fourth quarter, a little less than the 4.1 percent growth in the third quarter, the department said. But the end of inventory liquidations boosted the overall growth rate in the final quarter. Inventories, after adjusting for inflation, did not change in the fourth quarter after falling sharply in the previous three months.