A novel maize variety designated 1DLAX1066 and seed, plants and plant parts thereof are provided. Methods for producing a maize plant comprise crossing maize variety 1DLAX1066 with another maize plant are provided. Methods for producing a maize plant containing in its genetic material one or more traits introgressed into 1DLAX1066 through backcross conversion and/or transformation, and to the maize seed, plant and plant part produced thereby are provided. Hybrid maize seed, plants or plant parts are produced by crossing the variety 1DLAX1066 or a locus conversion of 1DLAX1066 with another maize variety.