Anorexia Nervosa (AN) is a severe and often enduring disorder characterized by restriction of food intake, low body weight, fear of weight gain, and distorted body image. Investigations on cognition performance in AN patients have yielded conflicting results. Using an established and sensitive computerized cognitive test battery, we aimed to assess core aspects of cognitive function, including attention span, information processing, reasoning, working and episodic memory, in AN patients and controls. Patients were recruited from the Danish Prospective Longitudinal all-comer inclusion study in Eating Disorders (PROLED). Included were 26 individuals with AN and 36 healthy volunteers (HV). All were tested with CogTrack (an online cognitive assessment system) at baseline, and AN patients were tested again at a follow-up time point after weight increase (n = 13). At baseline, AN patients showed faster reaction times in the attention tasks, as well as increased accuracy in grammatical reasoning compared to HV. There were no differences in cognitive function between AN patients and HV in the other cognitive domains measured (sustained attention, working and episodic memory, speed of retrieval, and speed of grammatical reasoning). No differences were visible in the AN sample between baseline and follow-up. Performance did not correlate with any clinical variables in the AN sample. These findings supplement results from other studies suggesting increased concentration and reasoning accuracy in patients suffering from AN, who showed increased performance in cognitive tasks despite their illness.