Trailing Eiking on the stage, French national champion Warren Barguil (Arkéa-Samsic) took the overall lead with a slim advantage of three seconds over Kazakhstan national champion Alexey Lutsenko (Astana).
"I was not at the front on the first slopes of Storheia, but I wasn't worried, because I knew the steepest sections still had to come," Eiking said. "I hoped for good legs, but in the end, they exceeded my expectations.
"I never felt in danger, and I easily joined the leaders. I started to think about the victory under the red flag after I saw how I easily made the jump.
"I wanted to start my effort with 300 meters to go, but when I saw riders coming back 600 meters before the finish, I took the lead and increased the rhythm bit by bit.
"I wasn't sure until the last 100 meters, because it was very steep and I was afraid to fall off my bike.
"Once I crossed the finish line, it was too good to be true I was happy to finish on the podium several times this year, but winning here is really special."
A five rider break got the 176.5km Queen stage from Sortland to the Storheia Summit started.
Erlend Blikra (Team Coop) was the last of the group to be reeled in as the climbers in the peloton began to exert their authority in the final five kilometres.
Eiking's Wanty-Gobert team-mate Loïc Vliegen was first to go followed by Barguil on a steep section 2.5km before the finish.
With 2km to go, race leader Mathieu van der Poel (Corendon-Circus) could not hold with the climbers and was forced to let Eiking, Barguil, Lutsenko and Hugo Houle (Astana) go.
Lilian Calmejane (Total Direct Energie) and Krists Neilands (Israel Cycling Academy) bridged across inside the steep last kilometre but with 400 metres to go, it was Eiking who surged as Barguil and Lutsenko drifted off his wheel.
They crossed the line in that order with Barguil taking the lead of the general classification from Van der Poel, with an advantage of three seconds over Lutsenko and 15 to Neilands.