The present study aims at investigating how eye contact established by a humanoid robot affects engagement in human-robot interaction (HRI). To this end, we combined explicit subjective evaluations with implicit measures, i.e. reaction times and eye tracking. More specifically, we employed a gaze cueing paradigm in HRI protocol involving the iCub robot. Critically, before moving its gaze, iCub either established eye contact or not with the user. We investigated the patterns of fixations of participants' gaze on the robot's face, joint attention and the subjective ratings of engagement as a function of eye contact or no eye contact. We found that eye contact affected implicit measures of engagement, i.e. longer fixation times on the robot's face during eye contact. Moreover, we showed that joint attention was elicited only when the robot established eye contact, whereas no joint attention occurred when it did not. On the contrary, explicit measures of engagement with the robot did not vary across conditions. Our results highlight the value of combining explicit with implicit measures in an HRI protocol in order to unveil underlying human cognitive mechanisms, which might be at stake during the interactions. These mechanisms could be crucial for establishing an effective and engaging HRI, and provide guidelines to the robotics community with respect to better robot design.