The entropy production by magnetohydrodynamic (MHD) flow and heat transport of non-Newtonian Carreau nanofluids through an irregular (converging/diverging) channel is addressed in this theoretical study. The second law of thermodynamics is employed to examine fluid flow, heat, and mass transport characteristics along with entropy generation arising within the system by incorporating the Buongiorno model for nanofluids. The Carreau nanofluids flowing across narrow and extending channels are modeled using the basic transport equations of continuity, momentum, energy, concentration, and entropy generation. Here, individual terms from the entropy transport equation are employed to study entropy accumulation and dissipation together within a flow. The transmuted equations are resolved via the Runge-Kutta Fehlberg technique and shooting quadrature in MATLAB software. The flow velocity, temperature, concentration, entropy generation rate, and Bejan profiles are evaluated and sketched graphically at various flow parameters. It's worth mentioning that the system's entropy generation grows along the diverse channel walls. Moreover, another important outcome of this analysis is that the rate of entropy formation increases as the Brinkmann number is estimated higher. This research could have substantial implications in high-temperature electronic devices, mechanical structures, heat exchangers, medical treatment, to mention a few.