Henrique scored 22 seconds into the second period to cut it to 3-1 after a pass to the low slot by Troy Terry. Henrique has a goal in four straight games.
"It was a basic, simple play that gave them some life and they ran with it," Blue Jackets coach Brad Larsen said. "This is a hard league; you can't give an inch. They've got some skill and you give them some room, and that's what happens. To give them that life early in the period, it puts you on your heels."
Fowler made it 3-2 at 13:07, and Simon Benoit tied it 3-3 at 15:02, using a screen by Lundestrom to score from near the blue line.
Zegras gave Anaheim a 4-3 lead at 5:07 of the third period with a shot from above the right circle that went off Merzlikins' glove.
"Obviously, we didn't get off to the start that we wanted to," Fowler said. "But at this point in the season, how tough things have gone for us, we still continue to fight and claw our way back in the game, and then a big third period. It's something we should be proud of."