Google has begun using billions of credit-card transaction records to prove that its online ads are prompting people to make purchases — even when they happen offline in brick-and-mortar stores, the company said Tuesday.

The advance allows Google to determine how many sales have been generated by digital ad campaigns, a goal that industry insiders have long described as “the holy grail” of online advertising. But the announcement also renewed long-standing privacy complaints about how the company uses personal information.