The 18-year-old Hendrickson outjumped Takanashi with a 106-meter effort on the normal hill in the first round and scored a total of 253.7 points. Her Japanese rival finished with 251.0.
Both jumped 103 meters in the second round.
"I can't even put it into words right now," said Hendrickson, of Park City, Utah. "I'm so happy. I had confidence on this hill and I knew I could have good jumps. But standing at the top of the first round, my heart was beating and everything was shaking and I didn't know if I could pull it off. But obviously I pulled together two good jumps."
Austria's Jacqueline Seifriedsberger was third with a total of 237.2 points. Her best effort of 104 meters came in the first round.
It is a positive sign for Hendrickson ahead of next year's Olympics, where women's ski jumping will feature for the first time.
"The first jump is really important for me mentally," Hendrickson said. "If I can have a good first jump I know I can have a good second jump. If I have a hard first jump sometimes I mentally shut down, so that was really important for me.
"Before the second jump it was snowing a bit up there so that blocked out the sound, which I really like. So I had no idea how far Sara went on the second round. I could hear the crowd roar but you never know with that. I just knew I had to do my jump and forget about the other stuff."
The 16-year-old Takanashi has won the last four World Cup events and also triumphed in the junior world championship last month, where Hendrickson finished sixth.
"Sarah is my strongest rival, my favorite jumper," Takanashi said. "The world championships are very different to World Cup events. All I can do is do my best."
Hendrickson won back-to-back World Cup events on the same hill at Val di Fiemme last season, setting the hill record with a 108-meter effort.
She dedicated the win to coach Paolo Bernardi, who comes from the region whose mother died this month.
"This is hometown for Paolo, where he was born and raised," Hendrickson said. "His dad basically built these ski jumps. I've had an amazing relationship with him over the past two years he's been coaching us, so to share this in his hometown is awesome. No words needed to be exchanged, just all hugs and happiness."
She is the second American to become the women's world champion after Lindsey Van won the inaugural event in 2009 in Liberec, Czech Republic. Van finished 16th on Friday.
Defending champion Daniela Iraschko of Austria was ruled out with an injury.