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.
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need
Kate Middleton recycles dress at movie premiere