The Cygnus region is a very bright and complex portion of the TeV sky , host to unidentified sources and a diffuse excess with respect to conventional cosmic-ray propagation models . Two of the brightest TeV sources , MGRO J2019+37 and MGRO J2031+41 , are analyzed using Milagro data with a new technique , and their emission is tested under two different spectral assumptions : a power law and a power law with an exponential cutoff . The new analysis technique is based on an energy estimator that uses the fraction of photomultiplier tubes in the observatory that detect the extensive air shower . The photon spectrum is measured in the range 1 to 200 TeV using the last 3 years of Milagro data ( 2005-2008 ) , with the detector in its final configuration . MGRO J2019+37 is detected with a significance of 12.3 standard deviations ( \sigma ) , and is better fit by a power law with an exponential cutoff than by a simple power law , with a probability > 98 % ( F-test ) . The best-fitting parameters for the power law with exponential cutoff model are a normalization at 10 TeV of 7 ^ { +5 } _ { -2 } \times 10 ^ { -10 } \mathrm { s ^ { -1 } \ > m ^ { -2 } \ > TeV ^ { -1 } } , a spectral index of 2.0 ^ { +0.5 } _ { -1.0 } and a cutoff energy of 29 ^ { +50 } _ { -16 } TeV . MGRO J2031+41 is detected with a significance of 7.3 \sigma , with no evidence of a cutoff . The best-fitting parameters for a power law are a normalization of 2.4 ^ { +0.6 } _ { -0.5 } \times 10 ^ { -10 } \mathrm { s ^ { -1 } \ > m ^ { -2 } \ > TeV ^ { -1 } } and a spectral index of 3.08 ^ { +0.19 } _ { -0.17 } . The overall flux is subject to an \sim 30 % systematic uncertainty . The systematic uncertainty on the power law indices is \sim 0.1 . A comparison with previous results from TeV J2032+4130 , MGRO J2031+41 and MGRO J2019+37 is also presented .