Kanepi, ranked No. 44 in the world, stormed to win the first set in 28 minutes, then carried a break win in the second set to victory.
Halep becomes the sixth No. 1-seeded woman to lose in the first round of a Grand Slam in the Open era. Angelique Kerber lost as a No. 1 seed in the first round of the 2017 French Open.