Rick Santorum won the Iowa caucuses Thursday — 16 days after the last vote was cast — when the state Republican Party said a final count showed him 34 votes ahead of Mitt Romney.

That was a shift from the preliminary results the party announced after the Jan. 3 caucuses, which showed the former Massachusetts governor winning Iowa by eight votes. Iowa Republican leaders said that they had still not received results from eight of the state’s 1,774 precincts.

The news does not alter the bottom line of the GOP primary race: the number of delegates that Santorum or Romney will receive at the national convention. For all their hype, the Iowa caucuses do not actually decide that.

But Santorum’s belated victory did seem likely to alter the pundits’ narrative of the Republican race — by demonstrating that the long-shot former senator from Pennsylvania did, indeed, have the ability to beat the front-runner.

At least once. By a little bit.

“The narrative that Governor Romney and the media have been touting of ‘inevitability’ has been destroyed,” Santorum communications director Hogan Gidley said in a news release. “Conservatives can now see and believe they don’t have to settle for Romney, the Establishment’s moderate candidate.”

Santorum’s squeaker victory means that Romney can no longer claim to be the only non-incumbent Republican candidate since 1976 to win both the Iowa caucuses and New Hampshire’s first-in-the-nation primary.

Romney, for his part, issued a statement calling the results “a virtual tie.”’

Santorum’s strange, belated victory also served to embarrass the Iowa GOP — which had to admit that it had misallocated some votes, and simply lost some others, in a razor’s-edge election where every vote mattered.

It also cast an unflattering light on the old-fashioned and convoluted system that the party uses to collect and count caucus votes.

“It should be like a fine Swiss watch,” said Iowa State political science professor Steffen Schmidt. “It’s really more like a sundial.” He said the system used by Iowa Democrats was not significantly better.

In fact, Iowa Republican leaders seemed to cast doubt on their own results, saying Thursday that it was hard to declare a “winner” without knowing what happened in those eight precincts. Matthew N. Strawn, the state party chairman, simply “congratulated” Santorum and Romney “on a hard-fought effort during the closest contest in caucus history.”

This is how the process is supposed to work:

On caucus night, local volunteers collect the votes of local Republicans. Often, this is done by asking voters to write their choices on small slips of paper, all in a uniform color to prevent fraud.

After the votes are counted up (and bad handwriting is puzzled out), local officials write each candidate’s total on something called a “Form E.” This form — but not the ballots themselves — is then sent to the state GOP.

Thursday’s final count came from these forms, which had to be submitted by Wednesday evening. The Des Moines Register, citing unidentified officials in the Iowa GOP, reported that in 131 precincts, the forms showed numbers different than those reported on caucus night.

But some Form E’s didn’t show up at all.

The state party found that it was missing results from eight precincts, spread across five counties. The eight precincts probably did not account for a huge number of votes — in the 2008 GOP caucuses, they had a total of 298 votes. But in this election, of course, 298 votes could easily have swung the outcome.

But now it is too late for the missing votes to count.

“It’s done,” said a party spokesman, who asked that his name not be used. About the missing votes, he said: “We never got ’em. We tried to track ’em down, and for whatever reason, we don’t have them.”

One of the missing forms was from the Geneva-Reeve precinct, which covers two small towns in Iowa’s midsection. Local volunteers in that precinct were supposed to return Form E to the county GOP chair, Karen Zander.

After the caucuses, Zander said she gathered paperwork that was sent to her from precincts across the county. She emptied the envelopes and sorted the loose papers into piles.

Then she bundled the Form E’s that she found together and sent them in. So how did that single one vanish? Zander said she doesn’t know: Did the volunteers at the caucus site lose it? Did she misplace it in the shuffle of paperwork?

“Now that the count is done, I don’t think that there’s anything we can do about it,” she said. “I’m going to take the blame for it, because I am the county chair. . . . If I blame one of [the volunteers], I’ll never get another precinct chair to volunteer.”

Staff writers Philip Rucker and Rosalind S. Helderman contributed to this report.