The Dane, seeded five, was getting overpowered as she trailed by a set and 3-5 to the powerful Kontaveit but she hung on grimly and turned the match around.
Her next task will be to break through to the quarter-finals for the first time, having fallen at the fourth round stage five times in the past eight years.
"I'm just happy I survived today," Wozniacki, who will face another stern test against American big-hitter Coco Vandeweghe on Monday, told reporters. "I'm not really too worried yet about Monday. I'm just going to enjoy having won today."
Kontaveit, one of the most dangerous unseeded players in the women's draw having won the Den Bosch grasscourt title in the build-up to Wimbledon, served for the match at 5-4 in the second set and was 30-0 ahead in that game.
She missed a routine backhand that would have given her three match points and Wozniacki levelled at 5-5.
World number 38 Kontaveit broke again to lead 6-5 but faltered on serve once more with the winning line so close.
She crumbled in the tiebreak and Wozniacki's experience paid off in the decider as the Dane moved through to a last-16 clash with hard-hitting 24th-seeded American.
"I felt like I wasn't playing my best in the beginning," the Dane said. "It definitely took me a while to get into it.
"I thought I played really well in the third set and at the end of the second set. I was pleased to just keep fighting. I felt her getting just a little bit nervous towards the end."
(Reporting by Martyn Herman; Editing by Ken Ferris)
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