Decays of $chi_{c1}$ to vector meson pairs $phiphi$, $omegaomega$ and $omegaphi$ are observed for the first time using $(106pm4)times 10^6$ $psip$ events accumulated at the BESIII detector at the BEPCII $e^+e^-$ collider. The branching fractions are measured to be $(4.4pm 0.3pm 0.5)times 10^{-4}$, $(6.0pm 0.3pm 0.7)times 10^{-4}$, and $(2.2pm 0.6pm 0.2)times 10^{-5}$, for $chi_{c1}to phiphi$, $omegaomega$, and $omegaphi$, respectively. The observation of $chi_{c1}$ decays into a pair of vector mesons $phiphi$, $omegaomega$ and $omegaphi$ indicates that the hadron helicity selection rule is significantly violated in $chi_{cJ}$ decays. In addition, the measurement of $chi_{cJ}to omegaphi$ gives the rate of doubly OZI-suppressed decay. Branching fractions for $chi_{c0}$ and $chi_{c2}$ decays into other vector meson pairs are also measured with improved precision.