The 14th seeded Slovakian advances to a semi-final showdown against China's Li Na, who edged Russia's Vera Zvonareva 6-4 7-5.
A day after upending third seed Martina Hingis in straight sets, the 2002 champion needed almost three hours to overcome some stubborn resistance from Peer, who consistently engaged Hantuchova in punishing rallies from the baseline.
The 23-year-old did well to overcome the disappointment of failing to serve out the match at 5-4 in the third set and rallied from a 5-3 deficit in the deciding tiebreaker.
Hantuchova drew level after Peer had committed two unforced errors, and a service winner gave her match point, which she took at the first attempt when her teenage opponent erred with a forehand.
"I think we both started to feel it in the middle of the third set because we had long rallies, and also the match yesterday with Martina was not easy," Hantuchova told reporters.
"I was running from one side to another, but it's a great feeling to be tired like that."
Even though she has ended the year ranked in the top 20 on four separate occasions, Hantuchova's only career title came here five years ago, but the player believes she has matured to the point where she is ready to challenge the top players.
"I'm understanding much better, if something goes wrong, it's part of the game, and it's okay, not to be perfect all the time," she said. "And just to try to figure out how to turn things around in my way and always try to stay positive."
Li was impressive from inside the baseline against the quick Zvonareva, finding the lines on the big points and tiring out her Russian opponent, who had upset world number one Maria Sharapova in the previous round.
Trailing 5-1 in the second set, the 25-year-old Li reeled off six straight games as Zvonareva lost depth on her shots and was pushed towards the back wall.
"After winning the first set, I was a little bit loose," the 12th seeded Li said. "At 5-1, I noticed the balls were coming back slower."
