Achieving a balance between the development of information and communication technologies (ICT) development and teaching competences has become essential. The rapid evolution of the Web has presented universities with the challenge of preparing today’s academic staff for the ICT of the future. The European Higher Education Area has adapted to ICT, proposing a conceptual and methodological change in teaching–learning processes in Spanish universities. In this new scenario, the education of students via the Web has become a key factor that requires higher education teachers to have new emotional competences. Although affections, feelings and emotions have been gaining relevance in society and scientific thought for more than a decade now, in the future, we will be dealing with a sensory emotive Web (Web 5.0) and, more than ever before, there will be a deep need for teachers to use and promote intra- and interpersonal emotional competences. In this respect, this paper suggests that the use of rationalistic methodology alone, in this era of sensory and emotional knowledge (present and future), is a mistake. Teaching staff need to develop emotional competences and transmit them to their students, in order to produce graduates who will be more adaptable to new socio-professional contexts.