Biron, making his fourth start of the season, stopped all 13 Capitals shots in the second period as the Rangers scored twice to grab a 3-1 lead.
Boyle's first goal of the year on a slap shot from above the left circle came midway through the frame, breaking a 1-1 tie, and Ryan Callahan tipped in a Rick Nash shot only 41 seconds later to make it 3-1.
Derek Stepan and Brad Richards also scored for New York, which improved to 4-1-0 in March.
Capitals goalie Braden Holtby started but was pulled in the second period after yielding three scores on 19 shots.
Michel Neuvirth relieved and stopped 10-of-11 tries the rest of the way.
Steve Olesky notched Washington's only goal.