Enhancing the sustainability, circular economy and use of by-products and renewable resources concerns a great number of researchers. Adding value to tropical vegetable resources and proposing a replacement for asbestos are challenges for chemists. Sugar cane bagasse is evaluated both as a mineral replacement and as a reinforcement in cementitious paste. In this matrix, ordinary Portland cement is partly replaced by natural pozzolan and bagasse ashes; a ternary binder is obtained. The pastes reinforced with vegetable fibers are prepared by the incorporation of untreated and pyrolyzed bagasse fibers (2–6 wt%) in this modified matrix. The thermal conductivity and the bending strength of the composite pastes, placed in various environments (curing chamber and water), are evaluated and compared to those of composite pastes made with commercial cement and bagasse fibers, exposed to identical aging conditions. The more relevant results are obtained in the curing chamber: the composite pastes prepared with the ternary matrix conduct less heat than the commercial binder composites at 28 d, and better bending strengths are obtained with the ternary binder composites at 28 and 90 d, particularly with the untreated bagasse fibers.