Jared McCann had a goal and an assist for Seattle (1-1-0). Joey Daccord made 34 saves.
Kirill Kaprizov had three points (one goal, two assists), and Mats Zuccarello and Matt Boldy each had a goal and an assist for Minnesota (1-0-1). Marc-Andre Fleury made 30 saves in his season debut.
It marked the start of Fleury's 21st NHL season, trailing only Martin Brodeur (22) for most seasons by a goalie.
Zuccarello scored his 200th career goal and made it 1-0 at 9:22 of the first period when Kaprizov, from behind the net, fed Zuccarello cutting through the slot.
Boldy increased the lead 2-0 at 1:06 of the second period with a lofted shot from the high slot that bounced off Daccord's pads.
Eberle responded 40 seconds later at 1:46, finding a loose puck down low to make it 2-1. It was his 300th career goal.
Eberle scored his second of the night to make it 2-2 at 8:15 with a backdoor deflection off an Andre Burakovsky shot.
Kaprizov put Minnesota on top 3-2 at 1:41 of the third period, tipping in a Marco Rossi shot, but Tye Kartye tied it 3-3 at 2:47 with a corner shot from the left circle.
Ryan Hartman broke the 3-3 tie at 12:33 after Boldy sent a behind-the-back pass to Hartman cutting through the slot.
With Daccord pulled for the 6-on-5 advantage on a delayed penalty, Ryker Evans found McCann atop the right circle for the one-timer and tying 4-4 goal at 16:34.