Condensation of non-Newtonian fluids is a less explored research field but has tremendous potential in the industry. We propose a mathematical theory to model the behaviour of non-Newtonian film condensation on vertical surfaces. The simple theory can contribute to understanding the operation of many engineering appliances using motor oils, polymeric liquids, and fuels. The well-known power-law model approximates the viscosity of condensate. We vary the power-law index (n) within 0.8≤n≤1.2 to encompass the behaviour of shear-thinning, Newtonian and shear-thickening films, while the flow consistency index (μ0) remains constant throughout the study. Since the film flow is gravity-driven, stream-wise advection is expected to dominate over cross-stream advection. We, however, establish the subtle role of cross-stream advection within the thin condensate film. We have identified a thinner film, a lower condensate drainage rate and a greater average heat transfer coefficient (hm) as an influence of the cross-stream advection. An exclusion of cross-stream advection transforms temperature distribution across the film from non-linear to linear. The linear temperature distribution is unrealistic for Ja≥0.5. We illustrate approximately 5% error incurred for neglecting cross-stream advection in estimating hm at Ja=0.81 and n=0.8. The error increases with a further decrease in n values or an increase in the Ja. The exclusion of cross-stream advection affects the film-thickness and average heat transfer disproportionately in shear-thinning and shear-thickening regimes. Finally, we propose mathematical expressions to estimate heat transfer coefficient and Nusselt number, which would be helpful for field engineers.