This paper presents a physically-based microstructural model for creep rupture at 600 °C for Grade 91 steel. The model includes constitutive equations that reflect various observed phenomena in Grade 91, and it is incorporated into a mesoscale finite element model with explicit geometry for the prior austenite grains and grain boundaries. Creep within the grains is represented using crystal plasticity for dislocation motion and recovery along with linear viscous diffusional creep for point defect diffusion. The grain boundary models include physics-based models for cavity growth and nucleation that accurately capture tertiary creep and creep rupture. Simulations of creep at 100 MPa are performed, and the contribution of each mechanism is analyzed. The overarching goal is to gain a mechanistic understanding of the material to improve the prediction of creep rupture for long service lives in elevated temperature operating conditions. The creep response of the material at different stress levels, stress states, and temperatures is studied in Part 2 of this paper in order to determine the implications of the simulations on high temperature design practice. Furthermore, the second part explores the effect of triaxial stress states on the creep response and finds a transition from notch-strengthening behavior at high stress to notch-weakening behavior at lower stresses.