The Kraken scored three goals in a span of 5:18 late in the second to take a 4-3 lead.
Tanev scored on a backdoor goal from the slot to make it 3-2 at 14:06 before Tolvanen scored on a one-timer from the top of the right circle to tie it 3-3 at 15:34.
Bjorkstrand made it 4-3 on a redirection from the slot at 19:24.
Vince Dunn scored on a slap shot 1:00 into the third period to push it to 5-3, and Wright connected from the slot to make it 6-3 at 11:13.
Miller pulled the Rangers within 6-4 on a shot from the left corner that hit Grubauer in the mask and went into the net at 12:27. Alexis Lafreniere cut it to 6-5 at 14:57 on a shot from the bottom of the right circle.
Yanni Gourde scored an empty-net goal at 18:14 for the 7-5 final.
"When we're at our best, we're playing fast, physical, and simple," Tanev said. "Everyone's on the same page. Those things all add up in a 60-minute game."
NOTES: Rangers goalie Igor Shesterkin, who signed an eight-year, $92 million contract Saturday, was unavailable after his wife went into labor earlier in the day. … With goalie Joey Daccord out for the game due to an illness, the Kraken signed Michael Matyas to an amateur tryout. The 32-year-old, who served as the backup, played with Alaska Anchorage in the NCAA for three seasons (2012-15). ... Wright extended his goal streak to three games and has six goals in his past eight games. ... The Rangers are 0-8-1 when trailing after the second period.