The officer was critically wounded after being stabbed three times in the stomach, but after surgery his condition was stable.
"The case is being investigated as an attempted homicide," local police said in a statement.
The suspect was a stateless Palestinian man born in 1990 who was about to be deported from the country. It is unclear how long he has spent in Denmark, or where the authorities planned to deport him to.
The centre, which lies around 20 kilometres (12 miles) northwest of Copenhagen, receives newly-arrived asylum seekers as well as those who have had their applications rejected and are due to leave the country.
Since Europe's refugee crisis began spilling into Denmark on September 6, at least 17,700 migrants have entered the country, according to police figures, but most have continued to neighbouring Sweden which has a more generous asylum policy.