Due to its high lethality among older people, the safety of nursing homes has been of central importance during the COVID-19 pandemic. With test procedures and vaccines becoming available at scale, nursing homes might relax prohibitory measures while controlling the spread of infections. By control we mean that each index case infects less than one other person on average. Here, we develop an agent-based epidemiological model for the spread of SARS-CoV-2 calibrated to Austrian nursing homes to identify optimal prevention strategies. We find that the effectiveness of mitigation testing depends critically on test turnover time (time until test result), the detection threshold of tests and mitigation testing frequencies. Under realistic conditions and in absence of vaccinations, we find that mitigation testing of employees only might be sufficient to control outbreaks if tests have low turnover times and detection thresholds. If vaccines that are 60% effective against high viral load and transmission are available, control is achieved if 80% or more of the residents are vaccinated, even without mitigation testing and if residents are allowed to have visitors. Since these results strongly depend on vaccine efficacy against infection, retention of testing infrastructures, regular testing and sequencing of virus genomes is advised to enable early identification of new variants of concern.