The Scottish Fire and Rescue Service was called out to the incident in Lochdale at about 01:45 on Tuesday.
The fire was put out but not before damage had been caused to the meter and cables inside.
PC Beth Williamson said it was was an "extremely dangerous act" which could have had serious consequences.
"The meter has been made safe and electricity restored," she added.
"We are keen to hear from anyone who may have seen or heard anything suspicious in or around the flat in the early hours of Tuesday morning."
Police have said the consequences of a suspicious fire at an electricity meter box at flats in Stranraer could have been "catastrophic".