New York sent the game into overtime thanks to John Hall's 31-yard field goal as time expired, then pinned the 49ers at their own four-metre line when safety Chris Hayes blocked a punt.
But Hearst took a hand-off, shed two tacklers and took off for the right sideline. He broke the tackles of the Jets' defensive backs, Kevin Williams and Otis Smith, and made it into the end zone just as linebacker Mo Lewis caught him from behind.
"That's going to go down in history now as one of the great, great runs of all-time," the San Francisco coach, Steve Mariucci, said. "I didn't even know he scored because of everybody going crazy down there."
The 96-yard run broke the previous team best of 89 yards, set in 1952 by Hugh McElhenny. It also helped Hearst finish with 187 yards - the fourth-highest total in team history.
In Philadelphia, Ricky Watters ignored the boos to rush for 67 yards on 16 carries, and Warren Moon threw three touchdown passes as the Seattle Seahawks kept their first clean sheet for 12 years with a 38-0 victory over the Eagles.
Watters played three seasons in Philadelphia, but during the off-season, he signed a four-year, $13m (pounds 7.8m) deal with the Seahawks. By the end of the game, the fans were booing the Eagles, who without Watters had no serious scoring threats. The only touchdown pass thrown by the quarterback Bobby Hoying was to the wrong team - to the Seahawks' cornerback, Shawn Springs, who returned it 42 yards for the game's first score.
The top two quarterback picks in the NFL draft this season, Peyton Manning and Ryan Leaf, made their debuts on Sunday with mixed results.
Manning passed for more than 300 yards but threw his first touchdown pass to the wrong team, finishing with three interceptions as the Indianapolis Colts fell to the Miami Dolphins 24-15. "I think the entire team and I will learn a lot from this game," Manning said. "If you turn the ball over you're not going to win."
The much-ballyhooed quarterback showdown between Manning, the first pick in the 1998 draft, and Miami's Dan Marino, the NFL's most prolific passer, never quite materialised, as Marino completed 13 of 24 passes for just 135 yards.
In San Diego, Doug Flutie's first NFL game in almost a decade upstaged Leaf's debut and nearly resulted in a storybook comeback, but Buffalo's Steve Christie missed a 39-yard field goal with seven seconds left and the Chargers held on to defeat the Bills 16-14. The second selection in the 1998 draft, Leaf completed 16 of 31 passes for 192 yards and a touchdown. He also threw two interceptions. "He made some plays, he missed some things and he'll just get better and better," the Chargers' coach, Kevin Gilbride, said.
Chicago managed a 24-23 home win over Jacksonville Jaguars in a game that featured five lead changes. In Kansas City, the Chiefs routed Oakland Raiders 28-8, meaning Kansas have now beaten Oakland in 16 of their last 18 games. Elsewhere, Cincinnati Bengals crashed to a 23-14 home defeat by the Tennessee Oilers while Pittsburgh won 20-13 at Baltimore.