Aho, who had an assist in the second period, became the first player in Hurricanes history with a 10-game point streak (four goals, 11 assists) to begin the season and the second in Hurricanes/Hartford Whalers history. Ron Francis began the season with a point in the first 11 games of the 1984-85 season with the Whalers.
Suomela scored his first NHL goal to give the Sharks a 1-0 lead at 6:37 of the first period.
San Jose took a 2-0 lead at 8:16 on Meier's power-play goal. Mrazek stopped a one-timer from Brent Burns, but Meier was alone in front for the rebound.
"I walked out at the end of the first period, where we could have been up 4-0, and we never won another race or a battle the rest of the night," Sharks coach Peter DeBoer said. "I guess the lesson in that is the NHL still plays 60-minute games, not 20-minute games."
After getting outshot 11-8 in first period, the Hurricanes quickly adjusted.
"[Brind'Amour] came in here and he was positive," Warren Foegele said. "We were sloppy. We believe in each other. You just look at each other like, 'I know we can come back,' and that's what happened."
Hamilton scored his first goal with the Hurricanes to make it 2-1 at 1:12 of the second period. His one-timer redirected off the stick of Burns and under the crossbar.
Hertl made it 3-1 when he took a pass from Logan Couture and finished for a power-play goal at 5:47.
McGinn's first goal of the season made it 3-2 at 10:07 of the second.
Aho picked off an errant cross-ice pass from Evander Kane and fed Teravainen, who scored on a wrist shot to tie the game 3-3 at 15:14.
"It was kind of (wide) open both ways and we didn't have a good start," Teravainen said. "But that's what good teams do. They might have a bad period, but they come right back at it. It's good to see that we can do that."