Vernon Davis’s greatest professional moment came attached to a bone-rattling collision with New Orleans Saints safety Roman Harper. So forgive the San Francisco 49ers tight end if he seems confused about how long he’s been dreaming of such a dramatic moment.

“Since ‘The Catch,’ ” said Davis, who attended Dunbar High in the District and the University of Maryland.

Considering that Dwight Clark made his dynasty-launching reception in 1982 and Davis was born in 1984, that seems somewhat improbable.

But the sentiment is appreciated. Davis’s 14-yard touchdown catch on a laser-like throw from quarterback Alex Smith with just nine seconds to play sealed a a breathtaking 36-32 victory over the favored Saints and put the 49ers in the NFC championship for the first time in 14 years.

Davis’s touchdown, his second of the game, capped a remarkable sequence of four lead-changing touchdowns, two by each team, in the game’s final 4 minutes 2 seconds.

First the Saints, who trailed 17-0 early in the second quarter, took their first lead, 24-23, on Darren Sproles’s 44-yard catch-and-run. But Smith, a playoff rookie and the target of derision for much of his seven years with the 49ers, answered on the next drive. He found Davis down the sideline for 37 yards, with a perfect over-the-shoulder pass. Three plays later, Smith rolled around the left end and ran 28 yards for a touchdown and the lead.

Brees, who threw two interceptions but completed 40 of 63 passes for 462 yards and four touchdowns, needed just four plays to regain the lead with a 66-yard touchdown pass to tight end Jimmy Graham. The Saints led 32-29.

With 1:37 to play, the 49ers’ offense got the ball back. Another deep pass to Davis got the 49ers in position to tie the score with a field goal. But the 49ers didn’t play it safe. They dialed up “Vernon Post,” a play they had been running in practice from the right side all week, but decided to run it from the left.

“I know there was a ‘Catch,’ ” said Jim Harbaugh, the coach who turned a 6-10 afterthought into an NFC powerhouse. “I don’t know what you’re going to call this one.”

Davis quickly entitled it “The Grab.”

The 49ers advance to play the winner of Sunday’s Packers-Giants game. If the Giants defeat the reigning Super Bowl champions, the 49ers will host the NFC championship next Sunday at Candlestick Park.

After eight seasons out of the playoffs, many spent as a league laughingstock, it’s no wonder Davis was overcome with emotion after his accomplishment.

“History was going through my mind,” said Davis, who finished Saturday’s game with seven receptions for 180 yards and two touchdowns. “Us against ‘No.’ Us against ‘Can’t.’ It was a very emotional game.”

Davis was also emotional for his longtime quarterback, Smith. So were all of Smith’s teammates.

“It’s fitting for all the stuff he’s been through,” tackle Joe Staley said. “To go toe to toe with Drew Brees and the New Orleans Saints.”

Saturday’s outcome was an unlikely and unpredictable turn of events in the 49ers’ unlikely and unpredictable season.

The teams came into the game from opposite ends of the football universe. The Saints’ goal was to accelerate, driving their record-setting offense through opponents. The Saints had won nine games in a row and were coming off a playoff victory over Detroit in which they piled up a playoff-record 626 yards.

The 49ers are designed as a brake — stopping other teams with their hard-hitting, opportunistic defense. That’s how they won most of their 13 games this season, capitalizing on others’ mistakes with a league-best turnover ratio of plus-28. Any chance of defeating the Saints was expected to start and end with the defense.

For much of the game on Saturday, that script worked to perfection. The 49ers rode their defense, piling up five Saints turnovers and opening up a 17-point lead.

The Saints had never won a playoff game in an opponent’s stadium, losing last year to the Seattle Seahawks. And they looked like the fresh air was bothering them again. But the Saints are one team that doesn’t find a 17-0 deficit daunting, and Brees coolly mounted comeback after comeback.

That’s when the doubts about the 49ers and their offensive abilities resurfaced.

But Smith and Davis, two of the players who made up the core of the 49ers through the hard times, connected in their biggest moment. The final touchdown reception — in the opposite end zone from where Clark made history — may ultimately go down as the beginning of a new era in 49ers history.

“Along the way, there’s been a lot of doubt,” Davis said.

Late Saturday afternoon, there was no doubt at all: the 49ers are back.