Population testing remains central to COVID-19 control and surveillance, with countries increasingly using antigen tests rather than molecular tests. Here we describe a SARS-CoV-2 variant that escapes N antigen tests due to multiple disruptive amino-acid substitutions in the N protein. By fitting a multistrain compartmental model to genomic and epidemiological data, we show that widespread antigen testing in the Italian region of Veneto favored the undetected spread of the antigen-escape variant compared to the rest of Italy. We highlight novel limitations of widespread antigen testing in the absence of molecular testing for diagnostic or confirmatory purposes. Critically, in the presence of a variant that escapes antigen testing, following up a proportion of negative antigen tests with a molecular test is the optimal testing strategy. Together, these findings highlight the importance of retaining molecular testing for surveillance purposes, also in contexts where the use of antigen tests is widespread.