The Vienna woman answered her telephone a few weeks back and was delighted to learn she was the lucky winner of a free vacation trip for two to San Francisco. Her name had been drawn at random in an advertising campaign, she was told, and all the caller needed now was a credit card number to verify her identity.

Preferably VISA or Master Charge, the caller added.

The woman was happy to comply. But last month, instead of airplane tickets to the coast, her mail box contained a credit card bill for hundreds of dollars worth of merchandise she had never ordered or received.

Telephone offers such as this one are the newest examples of a credit card scam that is sweeping the country, according to banking and consumer affairs officials. The schemes devised to trick credit card holders into disclosing their account numbers have worked so well in the Washington area that banks have begun warning their customers never to give out credit card numbers to unsolicited callers.

"The public just doesn't seem to realize that this little piece of plastic and those numbers on it are a valuable tool in the hands of crooks," warns John Carney, supervisor for fraud investigations at First Virginia Bank. "It's an industry-wide problem."

Banks and consumer groups report a marked increase in fraudulent credit card transactions, with estimated losses totaling hundreds of thousands of dollars. At a recent conference, Virginia and Maryland members of the International Association of Credit Card Investigators compared notes with law enforcement officials on the various types of ruses being used to obtain credit card numbers.

And the numbers, given the proliferation of mail order businesses and over-the-phone billing accounts, are almost as good as having the cards themselves.

Armed with a credit card number, people can charge airline tickets, appliances, theater tickets, stereo equipment, clothes and numberous other goods. And, since the holder still has the card in his or her possession, it is usually weeks before the fraudulent transactions are discovered.

In nearly all cases documented by banks, credit card fraud can be traced to drug addicts who use the cards to buy merchandise, which they then sell for cash.

"The crooks are figuring out everything they can to get the credit card numbers -- there's so much you can do with them," said Bill Cooper, vice president for United Virginia Bank's VISA card division.

The caller frequently uses a temporary or phony address when arranging to have items delivered. The goods are then claimed in front of the building on a prearranged delivery date or pickedup at the delivery office later with the excuse that there has been an address mix-up.

The tricky part of the scam, says Copper, is getting the credit card number.Yet what seems like a suspicious-sounding scheme in hindsight, he adds, "gets accepted as a perfectly ligitimate call" by the people who are caught off guard.

The prize or sweepstakes ruse usually works best because cardholders are so excited about their seeming good fortune. Victims are often told they will win if their credit card number matches one of those on the caller's list.

At other times, the caller may pose as a bank or credit card company employe who wants to "double check" an account or transaction. In the course of the conversation -- and maybe the caller also offers to raise the card holder's credit limit -- the customer ends up disclosing not only his card number but also the amount of charges he has made on the card so far.

"We've had so darn many of these," said First Virginia Bank's Carney. "Two years ago this type of complaint was almost nonexistent, but now we're getting them daily."

The problem is compounded, he said, because President Carter's credit tightening has made the cards "a lot harder to get" -- and thus more valuable. Also, by the time the card holder and the bank realize what has happened, it's almost the too late.

"There was this older couple in Oakton who gave out their account number and credit limit after being told by a caller it was for a survey," Carney said, citing a recent case. "The caller charged $570 worth of stereo equipment, had it sent to a District address and was there waiting outside the building when it was delivered. But we didn't hear about it for 60 days." t

Under federal law, card holders are responsible only for the first $500 of a fraudulent credit card charge, said Carney, "so this means the banks are out plenty."

The banking industry spends so much time investigating fraudulent card cases, he said, "that we (First Virginia) are getting ready to establish a procedure whereby you won't get a card again if you lose it or give out your number more than Twice."

Peter Drymalski, director of the Prince William County's Office of Consumer Affairs, recently alerted county residents to the dangers of giving out their card numbers over the phone.

"Sometimes the customer doesn't even remember giving the card number out," he said. "You might not think when someone starts asking a lot of questions that seem innocent. You might not think, but you should."