Catch and relief

Move over, Joe Montana to Dwight Clark.

Scoot aside, Steve Young to Terrell Owens.

Make room for The Catch 3.0.

San Francisco’s Alex Smith fired a 14-yard touchdown pass to Vernon Davis with nine seconds remaining Saturday, lifting the San Francisco 49ers to a 36-32 victory over the New Orleans Saints in a wild divisional playoff game at Candlestick Park.

A defensive gem of a game transformed into a shootout in the final minutes, with a series of rapid-fire touchdowns -- four in the final 4 minutes 2 seconds.

None was bigger than the last, when Smith dropped back quickly and fired a dart to Davis at the goal line in the middle of the field, unleashing the biggest 49ers celebration in more than a decade.

“I got the window and I cut it loose,” Smith said. “Vernon made a great play, in traffic, getting hit as he catches it. He deserves all the credit.”

There were plenty of backslaps to go around for a team making its first playoff appearance since 2002, a franchise with a new coach and downtrodden quarterback, coming off a 6-10 season. The 49ers are one win away from the Super Bowl and will play the winner of Sunday’s game between the New York Giants and Green Bay Packers for the right to represent the NFC on the biggest stage.

“Live or die, and we live, and we move on in spectacular fashion,” 49ers Coach Jim Harbaugh said. “I’m very proud of our football team.”

The Saints were favored by 31/2 points, the first visiting team to be favored in a divisional game since the 1996 season, when Dallas got the nod over host Carolina. (Like the Saints, the Cowboys lost.)

Saints Coach Sean Payton called Saturday’s thriller “kind of an unbelievable game, the way it went back and forth.”

“Obviously, it’s a disappointing game to lose, and we recognize the finality of it,” said Payton, whose team had won nine in a row. “To go through the momentum swings that we had, and we had our opportunities. Again, we credit San Francisco for making one more play than us to win a tough game.”

Perhaps the seeds of victory were planted Friday night at the hotel where the 49ers traditionally stay before home games.

“Last night in our highlight video we showed some clips of all the history-making playoff games,” 49ers tackle Joe Staley said. “At the end of the video they said, ‘History’s made in the playoffs,’ and I felt like today was one of those ESPN Classic games. You’ll see it on tomorrow.”

Smith’s final pass capped a dizzying flurry of touchdowns that started when New Orleans running back Darren Sproles caught a short pass and weaved his way 44 yards to the end zone with 4:02 left. That gave the Saints their first lead of the game, 24-23, and looked like it might have been enough in a game so dominated by defensive play.

But the 49ers weren’t done. Not even close. They answered with an 80-yard drive that consumed two minutes to move into field-goal range. They had third and three from the Saints’ 23 with 2:18 left when they were flagged for having 12 men in the huddle.

Pushed back five yards, the 49ers surprised the Saints with a keeper around the left side, with key blocks from Staley and receiver Kyle Williams that sprung Smith. The quarterback dashed untouched down the sideline for a 28-yard touchdown with 2:11 left.

Just before that play, Smith walked over and spoke to Harbaugh and offensive coordinator Greg Roman.

“Coach Roman as we were talking kind of mentioned that one,” Smith said. “Him and Coach Harbaugh were kind of debating that, and I loved [that play call], so I kind of jumped on it. . . . I like that QB run stuff. It adds a little dimension for us.”

So with a little more than two minutes to play, the 49ers had all the momentum, and -- after their conversion run failed -- a 29-24 lead.

The Saints had an answer, and a quick one. They moved 88 yards in four plays, reclaiming the lead, 32-29, with a 66-yard pass from Drew Brees to Jimmy Graham with 1:37 to play, and a two-point conversion pass to Sproles. It was a punch to the gut of everyone wearing red -- and there was a sea of them -- in the crowd of 69,732.

But that set the stage for history, a seven-play, 85-yard drive that will be burned into the minds of many who witnessed it. Davis, who made the winning catch, made another huge play on that drive: a 47-yard gain with 31 seconds remaining that put the 49ers in field-goal range to tie.

They never needed to attempt that kick.

Davis’ touchdown grab rekindled memories of other huge receptions by 49ers on their home turf -- “The Catch” by Clark to beat Dallas in the NFC title game after the 1981 season, and “The Catch II” by Owens to beat Green Bay in a wild-card game in the 1998 season.

“I know there was The Catch,” Harbaugh said. “I don’t know what you’re going to call this -- The Throw. The Throw and Catch.”

Whichever, it was the play that launched a million camera clicks and cellphone flashes. People know history when they see it.

“This does feel like a historic game, but this feels like a historic season for us,” 49ers safety Donte Whitner said. “From the way we started, even losing to the Dallas Cowboys in overtime, coming back and playing a lot of the great quarterbacks in the NFL and beating them, it’s been a historic year all year.

“Not giving up rushing touchdowns. All the things we’ve been able to do feels historic. This game today was nothing short of it.”

Wild scoring spree at the end of the game aside, San Francisco’s defense and special-teams units played a major role in the outcome. The 49ers forced five turnovers -- three fumbles and two interceptions -- and crimped the Saints’ running game, limiting New Orleans to 37 yards in 14 carries.

Although the Saints are a passing team, they rely on the run to set up those plays. Their ground game absorbed a big blow when running back Pierre Thomas sustained a concussion on the opening series and did not return.

Brees was masterful at times, completing 40 of 63 passes for 462 yards with four touchdowns, but was sacked three times and harassed throughout.

“Nobody expected us to win, period,” 49ers cornerback Carlos Rogers said. “Nobody expected the game to be close. Nobody expected with 1:37 left that our offense was going to score. But they did.”

And Smith was the unlikely architect -- though not unlikely, it seems, in the eyes of his teammates.

“I think [people] learned that he’s a winner and has tremendous heart,” Whitner said. “He can win big football games. He beat [Pittsburgh’s] Ben Roethlisberger on Monday Night Football. He beat [the Giants’] Eli Manning in a big game. He beat Drew Brees. He beat [Detroit’s] Matthew Stafford. I think it’s time to start mentioning him as a good quarterback in the National Football League.”

As Whitner was talking, Smith was pulling on a blue, short-sleeved work shirt, the one with the “Alex” label on the pocket -- a personalized gift given to the players from Harbaugh to remind them they’re a blue-collar group.

This wasn’t Alex the quarterback. This was Alex the mechanic. Ready to work another day.

sam.farmer@latimes.com

twitter.com/latimesfarmer

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(BEGIN TEXT OF INFOBOX)

Nothing finer

There were four touchdowns and four lead changes in the last 4 minutes 2 seconds at Candlestick Park:

*--* TIME PLAY 4:02 New Orleans: Darren Sproles 44 pass from Drew Brees (Kasay PAT).

Saints 24, 49ers 23 2:11 San Francisco: Alex Smith 28 run (run failed).

49ers 29, Saints 24 1:37 New Orleans: Jimmy Graham 66 pass from Brees (Sproles pass from Brees).

Saints 32, 49ers 29 0:09 San Francisco: Vernon Davis 14 pass from Smith (Akers PAT).

49ers 36, Saints 32 *--*