Decays of the chi_{cJ} states (J=0, 1, 2) to Lambda Lambda bar pi^{+} pi^{-}, including processes with intermediate Sigma(1385), are studied through the E1 transition psi-->gamma chi_{cJ} using 106 million psi events collected with the BESIII detector at BEPCII. This is the first observation of chi_{cJ} decays to the final state LambdaLambda bar pi^{+}pi^{-}. The branching ratio of the intermediate process chi_{cJ}-->Sigma(1385)^{+/-} bar{Sigma}(1385)^{-/+} is also measured for the first time, and the results agree with the theoretical predictions based on the color-octet effect.