Rookie Ronnie Brown rushed for 132 yards for Miami, while Carolina lost despite Smith's 11 catches for 170 yards and three scores, including a 53-yard touchdown midway through the fourth quarter to tie the game at 24.
The Panthers were driving for a potential winning score on their next possession when Schulters stepped in front of a pass intended for Keary Colbert and returned the interception 37 yards to the 25. Five plays, later, Mare kicked a 32-yard field goal with 4 seconds left.
"It doesn't get any lower than this," Delhomme said. "There was no doubt we were going to kick the field goal and win the game. You don't want to have to look your other players in the eyes, because you let them down."
Miami improved to 2-1 under new coach Nick Saban. Heading into a bye week, the Dolphins are halfway to last year's win total.
"We're not the worst team in the league like some people make us out to be," Schulters said. "We believe in ourselves and our abilities, and we like to stick it to people who don't believe in us."
The Panthers, who lost seven of their first eight games last season, are off to another slow start at 1-2.
"Back to the drawing board," coach John Fox said. "We're very disappointed."
Brown scored his first NFL touchdown on a 1-yard run, and his 58-yard dash set up a field goal. Playing much of the second half with a tight hamstring, he still finished with 23 carries and became the first Miami rookie in six years to rush for 100 yards.
"Especially against a defense like that, it feels great," Brown said.
Teammate Gus Frerotte threw touchdown passes of 18 yards to Randy McMichael and 42 yards to Chris Chambers, who made a one-handed grab at the goal line. Chambers' reception was initially spotted at the 1½-yard line, and the Dolphins were awarded the touchdown following a replay review.
Delhomme threw for 285 yards and one interception -- on his final pass, which came with the Dolphins in man-to-man coverage and blitzing. When the tight end stayed in to block, Schulters left him and cut in front of Colbert running a slant.
"I didn't see him," Delhomme said.
"All I had to do was catch the ball and run with it," Schulters said.
Carolina's offense was mostly Smith, who tied team records for receptions and touchdown catches. He traded taunts and shoves for much of the game with Pro Bowl cornerback Sam Madison, then beat him with an inside move for the tying score in the fourth quarter.
"Steve Smith put us in position to win, but unfortunately we didn't finish," Fox said.
Smith also fumbled attempting a fair catch on a punt to set up Miami's second touchdown. That started a flurry of three TDs by the teams in the final 5:44 of the first half, leaving the Dolphins ahead 21-17.
"They hit us with some big plays, and that's uncharacteristic of our team," Carolina linebacker Dan Morgan said.
One came with the Dolphins pinned at their 3 early in the fourth quarter. Brown found an opening up the middle for his big gain, which led to Mare's 27-yard field goal.
The second overall pick in the draft, Brown managed only 92 yards rushing in his first two NFL games, but after the long run he had fans chanting "Ron-nie, Ron-nie!"
Sunny, 85-degree weather likely helped the Dolphins, who are accustomed to the sweltering subtropical climate. They improved to 43-8 at home in August and September since the NFL merger in 1970.
Saban said Miami's success so far isn't necessarily seasonal.
"If we do what we're supposed to do," he said, "we can beat a lot of teams."