Gourde scored on a tap-in 39 seconds into the second period for a 3-0 lead off a turnover.
"We came out flying," Cirelli said. "It was a good start. We did some good things throughout the game. Obviously, there are some things we still need to get a little bit better at but it's always nice to come out with a win."
Boston responded to get within 3-1 at 1:09 when Mittelstadt scored on a wrist shot from the slot that beat Johansson to the short side.
Holmberg stuffed in a rebound at 3:37 of the second for a 4-1 lead.
Harris scored his first goal with the Bruins, finishing a 2-on-1 at 5:31 of the second to cut the lead to 4-2.
Harris, who is from Haverhill, Massachusetts, signed with the Bruins as a free agent on July 1.
"It's something that I've dreamed about, it's something that I've seen thousands of times on TV but special especially to be able to do it here at home," Harris said.