CINCINNATI — With their rookie quarterback playing mistake-free, the New York Jets turned their surprising playoff appearance into a long-running production.

Mark Sanchez handled single-degree wind chills and the playoff pressure with no problems Saturday, throwing a touchdown pass on a rollout play, and the NFL’s top running game took it from there, setting up a 24-14 victory over the Cincinnati Bengals.

Sanchez went 12 of 15 for 182 yards, becoming the fourth rookie quarterback since 1950 to start and win a playoff game. Considered the Jets’ weakest link heading into the playoffs, he became their focal point.

“He just kept getting better as the season went along,” coach Rex Ryan said. “I told him this was his second season now. His first season is over. This is the playoffs.”

Third-round pick Shonn Greene ran for 135 yards, including a 39-yard touchdown, as the Jets’ rookies made the pivotal plays under the biggest pressure.

The Jets (10-7) got their first playoff victory since 2004 and pulled off a rare back-to-back sweep of the Bengals (10-7), who got a team playoff-record 169 yards from Cedric Benson but little else. Cincinnati’s streak without a playoff win reached 19 years and counting.

The Jets?

“They might have a chance to make a move,” Bengals quarterback Carson Palmer said. “Their defense is that good.”

Cincinnati went to the Meadowlands six days earlier and got turned into road kill. The Jets ran for 257 yards, and the Bengals managed a total of 72 yards, with Chad Ochocinco getting shut out.

Little changed the second time around.

“This was a great team effort,” said Ryan, who won in his playoff debut as a head coach. “We’re a good football team. If people don’t believe that, they soon will.”

No one should count the Jets out now, not the way their coach did two weeks ago. Ryan thought the Jets were out of contention following a 10-7 loss to the Falcons that was set up by Sanchez’s three interceptions. Then, everything lined up in their favor.

The Colts pulled their starters a week later, allowing the Jets to rally for a win, while four other playoff contenders lost. Then, the AFC North champs showed up at the Meadowlands last week and lost 37-0 with little at stake.

Ryan’s father, Buddy, was the defensive line coach for the ’69 Jets, who won the Super Bowl title that Broadway Joe Namath had guaranteed. These Jets came into the playoffs as an off-Broadway show, lacking a star quarterback who could deliver a win.

Until Saturday.

Sanchez looked like a playoff pro, joining Shaun King, Joe Flacco and Ben Roethlisberger as rookie quarterbacks to win postseason starts. His passer rating was a stratospheric 139.4.

At times, Sanchez found himself on the sideline soaking it all in.

“It just blows your mind,” he said. “It’s unbelievable. I hope I have this feeling next week.”

Sanchez was by far the lowest-ranked passer in the playoffs, throwing 20 interceptions in his rookie season — second-most in the league. The Bengals wanted to put the game in his hands. Playing without a glove on his passing hand in an 8-degree wind chill, the kid from Southern California handled it without a bobble.

Afterward, the team present a game ball to owner Woody Johnson, whose daughter, Casey, was found dead in her Los Angeles home on Monday.

The Jets had a scare at the outset when punter Steve Weatherford was ruled out because of dizziness and an elevated heartbeat. Kicker Jay Feely punted for the first time in his NFL career, averaging 31 yards on seven kicks. He also made 20-yard field goal with 5:47 to go that put Cincinnati too far behind.

“I was so happy that I was kicking a field goal (then) instead of punting,” Feely said. “That was great.”

The Jets pulled ahead 14-7 by halftime with two big plays off Sanchez’s hand. He faked a handoff and made a perfect pitchout to Greene, who needed only one block to find open space for a 39-yard touchdown run, the longest of his career.

In the second quarter, Sanchez caught the Bengals off-guard. He rolled to his right and found tight end Dustin Keller running uncovered beyond the secondary. The throw was perfect, and Keller kept his balance for the last 15 yards while safety Chinedum Ndukwe vainly tried to knock him out of bounds.

At halftime, Sanchez was 7 of 10 for 94 yards with a passer rating of 132.9. It could have been even better — Braylon Edwards let a pass slip through his hands in the end zone.

Sanchez led an eight-play, 75-yard drive that culminated in Thomas Jones’ 9-yard run for a 21-7 lead late in the third quarter. Benson broke a 47-yard touchdown run — the longest in Bengals playoff history — that got Cincinnati within a touchdown, but Sanchez and Greene turned it on again.

The game ended with a little more Jets serendipity. Shayne Graham, the Bengals’ franchise-tagged player, missed two field goals in the second half, including a 28-yarder with 3:49 to go that essentially sealed it.

N.Y. Jets 0 14 7 3 — 24

Cincinnati 7 0 0 7 — 14

First Quarter

Cin—Coles 11 pass from C.Palmer (Graham kick), 7:07.

Second Quarter

NYJ—Greene 39 run (Feely kick), 11:52.

NYJ—Keller 45 pass from Sanchez (Feely kick), 6:19.

Third Quarter

NYJ—Jones 9 run (Feely kick), 2:18.

Fourth Quarter

Cin—Benson 47 run (Graham kick), 11:04.

NYJ—FG Feely 20, 5:47.

A—63,686.

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NYJ Cin

First downs 15 18

Total Net Yards 353 281

Rushes-yards 41-171 22-171

Passing 182 110

Punt Returns 2-26 1-23

Kickoff Returns 3-56 5-128

Interceptions Ret. 1-20 0-0

Comp-Att-Int 12-15-0 18-36-1

Sacked-Yards Lost 0-0 3-36

Punts 7-31.4 4-48.0

Fumbles-Lost 0-0 2-1

Penalties-Yards 9-76 6-55

Time of Possession 33:02 26:58

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INDIVIDUAL STATISTICS

RUSHING—N.Y. Jets, Greene 21-135, Jones 15-34, B.Smith 2-4, Sanchez 3-(minus 2). Cincinnati, Benson 21-169, C.Palmer 1-2.

PASSING—N.Y. Jets, Sanchez 12-15-0-182. Cincinnati, C.Palmer 18-36-1-146.

RECEIVING—N.Y. Jets, Cotchery 6-67, Keller 3-99, Edwards 2-15, B.Smith 1-1. Cincinnati, Coles 6-48, Cosby 3-26, Benson 3-12, Ochocinco 2-28, Caldwell 2-25, Leonard 2-7.

MISSED FIELD GOALS—Cincinnati, Graham 35 (WL), 28 (WR).