The event is being played at Gothenburg, Sweden.
A sellout and hostile crowd watched the Bryans hold steady after being stretched to tie-breaker in the first set.
The Bryans improved to 12-1 in Davis Cup play.
After a first set took 59 minutes, they needed just 68 minutes to win the match.
"I think that was our best Davis Cup match by far," said Mike Bryan. "We played two great sets so we're very happy."
Sweden needs to win both reverse singles matches Sunday to advance, when Andy Roddick faces Thomas Johansson, and if necessary, there will be a winner-take-all match between James Blake and Joachim Johansson.
Sweden is 3-1 against the U.S. at home, including a 5-0 victory in Gothenburg 10 years ago.
Notable deaths of 2014 [PHOTOS]