The upstart Liberty had tied the game on Betnijah Laney's long 3-pointer with 2.7 seconds left, but the Mercury got the ball to Turner underneath the basket on the ensuing inbounds play, forcing Sami Whitcomb to foul with 0.4 seconds remaining.
Turner's first free throw rimmed out but she calmly made the second, moving the fifth-seeded Mercury into the second round. Phoenix will play another single-elimination game at fourth-seeded Seattle on Sunday.
Sabrina Ionescu's last-second heave didn't come close for the eighth-seeded Liberty, who at at 12-20 had the worst regular-season winning percentage of any playoff team in WNBA history.
Skylar Diggins-Smith led Phoenix with 22 points and Brittney Griner had 16 points, 10 rebounds and six assists. Turner had 12 points and nine rebounds.
Laney led New York with 25 points, Natasha Howard had 16 points and 10 rebounds, and Ionescu finished with 14 points and 11 assists.
Phoenix hit eight of its first 10 shots to race to an 11-point lead, but by the end of the first quarter the Liberty had cut the deficit to 24-23. New York surged ahead by as many as six before leading 41-37 at the half.
Cunningham kept the Mercury in the game with 12 points in the third quarter, but they couldn't shake the Liberty.
Mercury top Liberty after late drama to advance
Betnijah Laney ties the game for New York, but Phoenix's Brianna Turner gets fouled and hits one of her free throws in the Mercury's 83-82 win.