In 2020, the World Health Organisation (WHO) published a strategy to eliminate cervical cancer as a public health concern. In South Africa, despite having a national screening policy in place since 2000, diagnosed cervical cancer incidence has shown no signs of decline. We extend a previously developed individual-based model for human immunodeficiency virus (HIV) and human papillomavirus (HPV) infection to include progression to cervical cancer. The model accounts for future reductions in HIV incidence and prevalence and includes a detailed cervical cancer screening algorithm, based on individual-level data from the public health sector. We estimate the impact of the current prevention programme and alternative screening scenarios on cervical cancer incidence. The South African screening programme prevented 8600 (95%CI 4700-12 300) cervical cancer cases between 2000 and 2019. At current levels of prevention (status quo vaccination, screening, and treatment), age-standardised cervical cancer incidence will reduce from 49.4 per 100 000 women (95%CI 36.6-67.2) in 2020, to 12.0 per 100 000 women (95%CI 8.0-17.2) in 2120. Reaching WHO's prevention targets by 2030 could help South Africa reach elimination (at the 10/100 000 threshold) by 2077 (94% probability of elimination by 2120). Using new screening technologies could reduce incidence to 4.7 per 100 000 women (95%CI 2.8-6.7) in 2120 (44% probability of elimination at the 4/100 000 threshold). HPV vaccination and decreasing HIV prevalence will substantially reduce cervical cancer incidence in the long term, but improvements to South Africa's current screening strategy will be required to prevent cases in the short term. Switching to new screening technologies will have the greatest impact.