Implementation of the European Higher Education Area changed the assessment model in higher education towards a competence-based system, so that students are assessed by what they demonstrate they can do (competences). However, assessment of certain key competences, such as reading, ceases at higher education. The question remains whether reading competence continues to influence academic achievement at this level. We designed a reading comprehension test based on PISA questions and administered it to new-intake university students. The results were linked to students' grades in each first-year subject and their sociodemographic administrative data. The influence of reading competence and socioeconomic status on grades in all first-year subjects was analysed through structural equation models. The results show that reading competence at the start of university education has a direct influence on grades in all first-year subjects and an indirect influence through the admission grade. In conclusion, reading competence needs to be further developed and promoted even in the university context.