Tanisha Wright had 18 points for the Storm, including a tying 3 with 23.9 seconds left in the first overtime. Camille Little added 12 points and Tina Thompson had 10.
Rebekkah Brunson scored 22 points and Seimone Augustus had 20 for the defending champion Lynx.
Thompson opened the second extra period with a 3 that put the Storm ahead for good, 78-75. After Lindsay Whalen had a jumper for the Lynx, Bird hit a 3 that stretched Seattle's lead to four points with 2:24 left.
Brunson and Little exchanged baskets 48 seconds apart, and Bird's 3 with 42 seconds remaining sealed the Storm's win.
Game 3 is Tuesday night in Minneapolis. The winner will face Los Angeles in the conference finals.
Lauren Jackson hit a 3 from the right side for Seattle that tied the score at 70-all with 1.1 seconds left in regulation.
Taj McWilliams-Franklin made 1 of 2 two free throws with 11 seconds that put Minnesota up three, setting up Jackson's tying shot.
Jackson, who shot 4 for 17, was scoreless on 0-for-6 shooting in the first half and had all nine of her points in the third and fourth quarters. She fouled out with 40 seconds to go in the first overtime.
Monica Wright's jumper put the Lynx up 72-70 with 4:07 left in the first extra period, and Little tied with a layup 50 seconds later.
Brunson hit two free throws with 2:09 to go, and then made another with 40 seconds remaining in the period when Jackson fouled out to put the Lynx up three.
Wright then hit her tying 3, and Augustus missed a jumper with 2.1 seconds to go, sending the game to another extra period.
The Storm trailed 57-47 with 3 1/2 minutes remaining in the third quarter before using an 18-6 run to take a two-point lead on Little's layup with 3:12 left in regulation.
The Lynx ended a scoreless stretch of more than four minutes when Brunson converted a putback to tie the score at 65-all with 1:31 remaining. She made a jumper nearly a minute later and Whalen made two free throws with 17.6 to go to put the Lynx up 69-65.
Bird's jumper pulled the Storm within two with 12 seconds left.
The Lynx called a timeout and Tanisha Wright fouled McWilliams-Franklin on the ensuing inbounds play in Minnesota's offensive end.