Belarusian Azarenka, the beaten finalist at Flushing Meadows last year, took control after breaking the Slovak with a superb backhand to go 5-2 up in the first set.
Hantuchova, ranked 48th in the world, made a determined start to the second set though, holding her serve and then going 2-0 up, breaking thanks to well-placed volley.
But Azarenka, who has reached the quarter-finals or better in eight of the last 11 Grand Slams, hit back immediately with a break which rattled her opponent, who proceeded to make a series of unforced errors.
Azarenka had three break points in the fifth game but lost them all, only for Hantuchova to double-fault twice at deuce and gift her the game and a 3-2 lead.
The Belarusian had service struggles of her own and gave Hantuchova a glimmer of hope with a double-fault that gave up a break to make it 5-3, but again the Slovak was unable to capitalize and Azarenka ran out the comfortable winner.
"It was a little bit tricky out there with the wind and Daniela is the kind of player who doesn't give you much rhythm," said the former world number one.
"So you really have to try to make it happen in one or two shots in the beginning, so that's what I was trying to do."
It was a disjointed game in many respects, which Azarenka said hampered her ability to find her flow.
"There weren't many long rallies and that sometimes makes it more difficult to get into the match. But I felt like I was really good in the important moments," she added.
"In the first set, towards the end, I really raised my level to where I wanted to and that was the key to this match."
Hantuchova, who was in her first US Open Grand Slam quarter-final since 2002, knows she will need to find greater consistency and limit her errors if she is to go further next time.
"I didn't feel like I played my best tennis today, which is unfortunate because I had many chances and didn't hold my serves when I needed to," she told Reuters.
"Throughout the whole tournament my serve was working great but just not tonight.
"I am very disappointed with the performance, because I know what I can do and how I can play. I came into the match feeling confident, it just didn't work out today."
- Sports & Recreation