Taurasi had a personal playoff-best 34 points and DeWanna Bonner added 13 for the Mercury, who had the league's best record during the regular season at 29-5.
Candace Parker, who led the Sparks with 22 points, threw too high for Kristi Tolliver with 7.1 seconds left, giving Phoenix the ball with a 73-70 lead. Bonner hit her third and fourth straight foul shots down the stretch to help secure the win.
It was the sixth win of the season for Phoenix over Los Angeles, which faces elimination in the best-of-three series on Sunday.
The Sparks, who went 16-18 during the regular season, got two quick baskets to take a 68-65 lead with 2:57 to go, but struggled to score in the final minutes.
Bonner's foul shots with 18 seconds left put the Mercury up by three. Parker then answered with a driving shot in the lane with 11 seconds remaining, trimming Phoenix's lead to 71-70.
Armintie Herrington fouled Taurasi with 9.3 seconds left.
Parker scored seven points during a 9-0 fourth-quarter run that gave Los Angeles a 62-58 lead with 6:23 left.
Nneka Ogwumike had 12 points and six rebounds for the Sparks, and Tolliver finished with 13 points.
Phoenix star Brittney Griner was 2 for 11 from the field for six points. She also had seven rebounds and blocked two shots.
The Mercury made three of their first four shots of the second half on their way to a 54-44 lead, taking advantage of Los Angeles foul trouble. But Tolliver hit a 3-pointer and Parker found Tolliver with an outlet pass to cut the lead to 54-51 with 1:03 left.
Phoenix led 56-53 going into the fourth quarter. Taurasi, who became the first player in league history to lead the WNBA in points in one season and assists in another, was 10 for 15 from the field. She went 6 for 9 from beyond the arc.