World No.15 Pallikal and her Danish opponent battled it out for 76 minutes before the Indian scraped through with a 13-11, 14-16, 11-6, 8-11, 11-4 result at Ponterfact, about fifty miles from here.
The Indian said she was lucky to get over the line against 22nd-ranked Hansen in the first round match played on Tuesday night.
"I was not as patient as I could have been for most part of the match. She (Hansen) has improved a lot and I knew it would be a tough match," Pallikal told IANS.
"In the deciding fifth game, I told myself to stay patient and stay on there in the rallies. And that helped."
The action now shifts to the multi-purpose KC Stadium where the round of 16 matches will be staged in an outdoor glass court.
Pallikal next plays local favourite Laura Massaro on Thursday and she knows she will have to come up with something special against the number two seed. She will also have to adjust to the outdoor conditions.
"For starters, I would have to be a lot more patient than I was in this game. The venue is an outdoor one so the balls are going to be more dead, and I have to choose the right one," she added.
Pallikal last played Massaro at the 2012 Australian Open semi-finals where she lost after leading 9-5 in the decider.
India's Saurav Ghosal faces fourth seed Gregory Gaultier of France in the men's pre-quarterfinals later on Wednesday.
The British Open is the world's oldest tournament which makes a return to Yorkshire after almost 40 years.
British Open squash: Dipika Pallikal wins marathon match to reach pre-quarters
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