Bergeron cut the Carolina lead to 3-1 at 14:57 of the second period when he followed his own backhand and slipped the puck under Kochetkov.
Niederreiter scored after a goal-front scramble during a 5-on-3 to make it 4-1 lead with 1:08 left in the second period.
Bergeron redirected Charlie McAvoy's shot with his left skate to make it 4-2 at 12:21.
"One game at a time. That's all we can focus on now, it's game three," Bergeron said. "It's playoff hockey, you have some adversity. For us it's early. So, you know, back to game three."
The Bruins had a 6-on-4 after pulling Ullmark and going on the power play late in the third, but Kochetkov stopped Pastrnak on a back-door pass before making a save on McAvoy in the slot.
Niederreiter scored an empty-net goal with 41 seconds remaining for the 5-2 final.
NOTES: Kochetkov (22 years, 313 days) is the third-youngest goalie in Hurricanes/Hartford Whalers history to make his playoff debut, behind Kay Whitmore (21 years, 360 days) & Cam Ward (22 years, 54 days). … Boston defenseman Hampus Lindholm left the game late in the second period after a check from Svechnikov. Cassidy did not have an update on Lindholm. … Bergeron scored his 47th playoff goal to pass Phil Esposito for second on the Bruins all-time list (Cam Neely, 55). … Niederreiter has three goals in the first two games of the series, matching his total in 29 playoff games the past three seasons. … DeAngelo has four assists in the series. … Svechnikov had a career-high nine hits. … The Hurricanes are the second team in NHL history to have two different goalies each record his first playoff win in the first two games of a postseason (Detroit Red Wings, Ron Low and Jim Rutherford, 1978). ... Carolina has won five straight against Boston this season, outscoring its opponent 26-4. The Hurricanes defeated the Bruins 3-0 on Oct. 28, 7-1 on Jan. 18, and 6-0 on Feb. 10 in the regular season, and 5-1 in Game 1 on Monday.