Maya Moore scored 22 points and Rebekkah Brunson had 15 points and 13 rebounds for Minnesota, which won the WNBA title in 2011 and lost to Indiana in last year's finals.
Minnesota opens the Western Conference finals this week against the winner of the Los Angeles-Phoenix series, which will be decided on Monday.
Tina Thompson made a driving layup to put the Storm up 55-54 with 1:20 remaining, their first lead since late in the second quarter.
But Augustus, who had 10 points, responded with her jumper from the right side. Brunson hit two free throws with 11.6 second left for the final margin.
Noelle Quinn missed a driving layup in the final seconds for Seattle, which rebounded the miss but couldn't get off another shot in the game played at the Tacoma Dome because KeyArena in Seattle was unavailable due to a scheduling conflict with a Microcsoft event.
Thompson had 13 points and nine rebounds in the final game of her 17-year career. She announced before the season that she would retire at the end of the year. The 38-year-old forward is the WNBA's career scoring leader and won four league titles with Houston.
Tanisha Wright led Seattle with 16 points. Camille Little added 14 points as the Storm lost in the first round to the Lynx for the second straight year. Seattle made its league record 10th consecutive playoff appearance.
Moore scored seven in the third quarter to help the Lynx to a 42-39 lead. Wright scored six for the Storm.
Moore scored 12 as the teams were tied at 28 at halftime. The Storm outscored Minnesota 12-8 in the second quarter as the Lynx made only 4 of 15 shots from the field.
The Lynx beat the Storm 2-1 in last year's Western Conference semifinals.