Bob Turner, once considered a long shot to replace disgraced former Rep. Anthony D. Weiner, is rolling toward Tuesday’s special election with a rush of momentum rarely seen by a Republican in the New York City district.

Although Mr. Turner is expected to benefit from low voter turnout and President Obama’s low job-approval ratings, Democrats are banking on an influx of cash, a flood of TV ads, an army of campaign volunteers and a reliable party base to push their candidate, David Weprin, to victory.

“This is one of those places where the Democratic machine knows how to get out the vote, and Republicans just don’t have that in place,” said a senior House Democratic aide.

Republicans agree that winning the district, which includes parts of Brooklyn and Queens and where Democratic voters outnumber Republicans by more than a 3-to-1 margin, won’t be easy. But with an independent Siena College poll taken last week showing Mr. Turner with a 6-percentage-point lead — a result almost unimaginable earlier in the summer — the party is in good shape to capture the seat for the first time since 1921.

“We’re hopeful we’ll have a strong operation heading into Election Day, but it’s going to be tough, though, because Democrats have the political machine in Queens and New York City, and so that’s something that we’ll have to overcome,” said a senior House Republican aide...