KABUL, Afghanistan — After enduring months of Taliban attacks and days of security clampdowns, Afghans reveled Sunday in the apparent success of the weekend’s presidential election, as officials offered the first solid indications that the vote had far exceeded expectations.

Two senior officials from the Independent Election Commission said the authorities supervising the collection of ballots in tallying centers had counted between seven million and 7.5 million total ballots, indicating that about 60 percent of the 12 million eligible voters had taken part in the election. The officials spoke on the condition of anonymity because results will not be released for weeks.

At least some of the votes are expected to be disqualified for fraud, but if the numbers hold up, they will buttress anecdotal accounts of Afghans voting in large numbers on Saturday in the country’s first wide-open election, with at least three of the eight candidates considered contenders to replace President Hamid Karzai. Afghan and Western officials, including Nicholas Haysom, the United Nations’ top election official, had said turnout above 40 percent would be an excellent result.