The Estonian 32nd seed blew a set and 4-1 lead with two breaks to allow the 39th-ranked Spaniard back into the match.
She also stumbled when serving for the match at 5-4 in the third, as the experienced Suarez Navarro reached her third Australian Open last eight after 2hr 15min.
“Was tough today. She started really well,” said the Spaniard who has also reached the quarter-finals twice at Roland Garros and also at the US Open, but never progressed further.
“I always have to fight to the end,” she added. “I just tried to play my game and stay really focused. I just run and run. That’s it.”
Oozing confidence after despatching French Open champion Jelena Ostapenko in the third round, Kontaveit was on the front foot from the off.
Estonia’s Anett Kontaveit blew a set and 4-1 lead with two breaks to allow 39th-ranked Spaniard Carla Suarez Navarro to run out an eventual 4-6, 6-4, 8-6 winner in their fourth round match at the Australian Open
Suarez Navarro was ranked in the top 10 less than two years ago but has been on a downward slide and had no answer to the pace of Kontaveit’s early ground strokes.
The 22-year-old made the decisive break in the fifth game and secured a 5-3 lead with the standout shot of the first set, a superb running forehand pass that flew around the net post to bring the Rod Laver Arena crowd to their feet.
Kontaviet served out after 42 minutes and in the second, a bullet-like backhand gave her a second break of serve to 4-1 up with one foot in the quarter-final.
But 29-year-old Suarez Navarro, whose first Grand Slam quarter-final at Melbourne Park was nine years ago, drew on her experience and began to play a patient game, extending the rallies.
Nerves set in for Kontaveit as she neared her first Slam quarter-final and a forehand which had powered 15 winners in the first set alone faltered.
– Lifeline –
When she double-faulted on break point, Suarez Navarro had a lifeline at 4-2.
The Spaniard easily held for 4-3 and levelled the set as Kontaveit went wide with another misdirected forehand.
The momentum shift was completed when Suarez Navarro held again and Kontaveit had to serve to save the set at 4-5, having been 4-1 up.
Anxiety took over the young Estonian. A double fault and another forehand error helped Suarez Navarro create two set points.
Kontaviet regathered to save those but, when a second double fault brought up a third, Suarez Navarro’s stunning comeback sent the match went to a decider.
A ripping winner crosscourt brought up a break for Kontaveit to lead 5-4 but as in the second stanza nerves prevented her getting over the finish line and Suarez Navarro broke back.
At 6-7 Kontaveit’s capitulation was complete when she conceded match point on her serve, the Spaniard taking her chance when the Estonian went long.
The reward for the former world number six is a clash against either second seed Caroline Wozniacki of Denmark or Slovakia’s 19th seed Magdalena Rybarikova.
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