Latvian Ostapenko produced her usual brand of gung-ho tennis but Estonian Kontaveit weathered the storm to take her place in the fourth round in Melbourne for the first time.
Ostapenko, playing with a heavily bandaged thigh, dominated the second set but was unable to sustain her level.
The players traded breaks early in the deciding set but the 22-year-old Kontaveit outlasted her opponent in marathon seventh game, thumping a weak second serve for a backhand winner on her fifth break point to seize the advantage again.
After holding serve, 32nd seed Kontaveit sealed victory when an Ostapenko forehand thumped into the net.
Her reward is a fourth round tie against unseeded Spaniard Carla Suarez Navarro who got the better of Kaia Kanepi in three sets.
(Reporting by Martyn Herman; Editing by Toby Davis)
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