With mentor Martina Hingis in the stands, 17-year-old Bencic more than held her own against former world number one Jankovic, winning 7-6 (8-6) 6-3 to earn a place in the quarter finals.
Few gave Bencic a chance against her experienced opponent, and the Swiss sensation believed this allowed her to simply enjoy the moment under lights on Arthur Ashe.
"No, I definitely didn't feel any pressure because it was my first time on the Arthur Ashe in the night session. So I really just tried to enjoy it," she said in her post-match press conference.
"I think she [Jankovic] had all the pressure because she needed to win. I could just enjoy. I mean, I would have many more chances if I didn't win today, so..."
Bencic showed great nerve to save three set points in the opener, as Jankovic twice failed to serve out the set before eventually going down in a tie-break.
The situation was reminscent of Bencic's third-round match against Angelique Kerber and the world number 58 said the experience helped her on Sunday.
"It was similar like against Kerber. I turned that set also completely. Here I also had nothing to lose," Bencic said.
"I was also ready to fight in the second and maybe third set, but I just fought till the last point. In the tie-break I played really well."
Bencic, the youngest player since Hingis to make it through to the quarter-finals of the US Open, revealed the five-time grand slam champion had some words of advice before her match against Jankovic.
"She told me to hold the rallies. I have to be aggressive," Bencic said.
"I had to pay attention to her backhand down the line. So I really knew she was doing that a lot. Her backhand is just amazing, so I tried to push it to her forehand."
Bencic will now take on China's Peng Shuai in a bid to earn a place in the last four.