Brooklyn rents — especially for studios — are soaring as apartment hunters are finding cheaper alternatives to pricey Manhattan on the other side of the East River.

Median rents in Brooklyn jumped 10.1 percent, to $2,698 a month, compared with the same period last year, a new study found.

Manhattan rents are still higher, with a median of $3,195. But they rose at a modest 1.4 percent in October compared with last year — the lowest increase in more than a year.

“Brooklyn is catching up,” said analyst Jonathan Miller, of Miller Samuel Inc., who prepared the report for Prudential Douglas Elliman. “It’s become a less-expensive alternative.”

The more affluent neighborhoods in northern Brooklyn — including Williamsburg, Greenpoint, Brooklyn Heights and Cobble Hill — are in special demand.

In addition to “the affordability factor,” Brooklyn’s boom is being fueled by the very tight credit market — which makes renting more attractive than buying a home — and the improvement in the city’s economy in 2012, Miller said.

“The rental market is very responsive to changes in employment,” he added. “Studios tend to be the first to react.”

Studio apartments in Brooklyn have a median rent of $1,750, up 9.4 percent compared with rents a year ago. That’s influenced by an influx of newly-hired workers looking for entry-level housing.

Manhattan studios are still more expensive, at a median rent of $2,300. But that’s down 2.1 percent year over year.

Another feature of today’s rental market is that apartment dwellers are looking to move when they see their lease renewal with its rising rent bill. The number of new rentals rose 59 percent in Brooklyn and 41 percent in Manhattan compared with the same period in 2011.

“There’s more churn going on,” Miller said. “We’re seeing some consumer resistance to increases.”

Nevertheless, Manhattan’s vacancy rate continues to fall, down to 1.59 percent, from 2.32 percent a year ago.