In addition to the graves, paint was also used to mark the town's Åssiden Church, newspaper VG reports.
Police were made aware of the offence at 10:15am on Tuesday and are now looking for an adult man they think may be connected to the incident.
“The name of a company has been tagged on to all of the headstones, and the individual under our suspicion did something similar a couple of weeks ago,” unit leader Marianne Mørk of the Sør-Øst (South East) police district told VG.
Mørk said that the man in quesion had previously been turned down for a job with the company named in the paint tags.
Local newspaper Drammens Tidende was first to report the incident.
Polcie said that the type of paint used on the headstones on the church is likely to be expensive to remove. Numbers were also tagged on some of the headstones, according to the report.
“I have not seen such comprehensive damage of this kind before,” Mørk told VG.
The churchyard was closed while police carried out investigations but has since been reopened.
Åssiden Churchyard manager Stig Indregård spoke of his despair at the incident.
“It is awful that somebody has done this to a churchyard,” Indregård told Drammens Tidende.
“We must start be seeing what the extent of the damage is and how we can remove it, whether it can be removed on site or whether the gravestones must be removed to be repaired,” he added.