Kshama Sawant seized a narrow lead over longtime Seattle City Councilmember Richard Conlin on Tuesday, a week after an election night in which most assumed she had lost.

Sawant, who would be the council’s first socialist member in modern history, led by 41 votes — 79,751 to 79,710 — in updated vote totals.

The slim lead is stunning considering that election-night returns gave Sawant 46.1 percent and only a faint glimmer of hope.

Now, her momentum has put her in position to win or to force a recount.

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“The most important feeling here is not only confirming the trend toward our campaign,” said Sawant, arguing that those who mailed their ballots late were poorer, younger voters, “but in reality, this is confirming our analysis that there is a deep hunger among people not just in Seattle but everywhere in the United States for a deep political change.”

Conlin did not return telephone messages seeking comment Tuesday.

Election officials estimated there were about 62,000 ballots from King County left to be processed. The number for Seattle is close to one-third of that.

There were also about 9,700 ballots — again, countywide — with signatures that have been challenged. Both campaigns worked over the weekend to identify and help any supporters whose signatures were challenged.

The Sawant campaign said it had raised more than $8,000 since election night and had more than 100 volunteers knocking on doors over the weekend.

“That’s just the beginning,” political director Philip Locker said. “We intend to make sure all ballots are cured and rehabilitated.”

Conlin has been reaching out to supporters through email.

The efforts by both campaigns indicate the race could go on until results are certified Nov. 26 — or beyond.

A mandatory machine recount will be triggered if the vote difference is less than 2,000 votes and less than 0.5 percent of the total votes cast.

Either Sawant or Conlin could avoid that by winning roughly 54 percent of the remaining votes, according to a Seattle Times analysis.

At this point, that appears more possible for Sawant.

She won 57.4 percent of the 8,395 votes counted Tuesday. Overall, she has won 54 percent of the 77,173 votes counted since election night, when she was far behind.

Nick Licata, who is tied with Conlin for the longest tenure on the council — 16 years — said he has never seen a Seattle election turnaround like this one.

“Not even close,” Licata said. “This is historic.”

The only possible comparison in recent years, he said, is a 2011 Seattle School Board race in which challenger Sharon Peaslee came back to win after taking just 47.7 percent of the initial returns.

But despite Sawant’s momentum, Licata declined to speculate on who would ultimately prevail.

“If there’s anything this election has taught me, you can’t predict anything.”

Staff reporter Justin Mayo contributed to this report.

Brian M. Rosenthal: 206-464-3195 or brosenthal@seattletimes.com. On Twitter @brianmrosenthal