Teenager Bencic and Hingis, 35, who won her 12th Grand Slam doubles title in last weekend's Australian Open final alongside India's Sania Mirza, needed just over an hour to seal their 6-3, 6-2 win over Germany's Andrea Petkovic and Anna-Lena Groenefeld.
"I was cheering them on for the first four matches and then I had to go out there," said a delighted Hingis after the tie in Leipzig.
Bencic won three points at the weekend to secure a semi-final place where the Swiss will take on defending champions Czech Republic, who edged Romania 3-2.
The teenager said she was delighted to have the chance to play alongside Hingis for her country.
"She did all the great volleys and all the doubles stuff," said the 18-year-old, who is ranked 11th in the world for singles, but only 59th for doubles.
"I was just happy I could play with her. I'm not a doubles specialist."
Switzerland's captain Heinz Günthardt said the sky is the limit for his team with both Bencic and Hingis in top form.
"I have no idea where we can go but I'm excited to find out," he said.
"Martina is one of the best doubles players in the world right now and Belinda is very solid.
"This surface is good for Belinda, but we knew she had to play at the top of her game and she did."
Sunday's defeat was tough on Petkovic, who lost the opening rubber to Bencic on Saturday and was then stood down for Sunday's reverse singles when Annika Beck had levelled the tie at 2-2.
Beck's 7-5, 6-4 victory over Timea Bacsinszky had given the Germans brief hope before Hingis and Bencic sealed the Swiss win.
The 21-year-old Beck had rescued Germany after Angelique Kerber lost to teenager Bencic in Sunday's first singles rubber which had put the Swiss 2-1 up.
Just eight days after beating Serena Williams in the Australian Open final, Kerber had few answers as she lost 7-6 (7/4), 6-3 to the 18-year-old.
It had finished all square on the first day after Bencic, the world number 11, swept past Petkovic, 6-3, 6-4, then Kerber, ranked second in the world, got Germany back on track with a 6-1, 6-3 win over Bacsinszky.