The siblings were fourth after the short program, but scored 114.86 points in their free skate en route to an overall total of 192.59.
The duo is skating in their second Olympic Winter Games.
"It feels like gold,” Alex Shibutani said. “It's unbelievable. I am so proud of the way we fought through this week and the season. We are so emotional.
"I was really proud of how we skated today. We knew that regardless of what the result was going to be, we did everything that we could and have no regrets. We are really proud of each other and the result was amazing."
The United States has now won an ice dance medal in four consecutive Olympic Winter Games.
Madison Hubbell and Zachary Donohue finished fourth with 187.69 points. They earned 109.94 points in their free dance. Donohue experienced an error at the end of the program, which resulted in a deduction.
"It just wasn’t our best skate today,” Hubbell said. “We felt really focused during the performance, just a little slip here and there and we lost some really valuable points. In such a tough competition, there’s really no room for error.”
Madison Chock and Evan Bates also finished in the top 10, earning 175.58 points for a ninth- place finish. The couple was skating a solid program when they fell at the midpoint as they entered a spin.
"We were so close,” Chock said. “As soon as [we fell] we knew there was no way. It was pretty hard, right away to keep going. We owed it to ourselves and our choreographers and our team to keep going and put out the rest of the performance."
Canadians Tessa Virtue and Scott Moir won the gold medal while the French team of Gabriella Papadakis and Guillaume Cizeron earned the silver medal.
The ladies short program will get underway on Wednesday at 10 a.m. (Tuesday, 8 p.m. ET).
