The 56-year-old was sacked by Chester in April after a run of just one win in 10 put the club in a relegation fight.
Southport relieved Andy Bishop of his duties on Saturday with the side lying second from bottom in the table.
Ex-Kidderminster boss Burr takes over a side that has picked up four points from their first eight league matches of the season, suffering six defeats.

The collapse took place very close to the famous Ponte Vecchio, a medieval covered bridge over the Arno.
The hole is about 200m (650ft) in length and 7m (23ft) across.
Firefighters believe the chasm has been created by the failure of a large water pipe beneath the surface.
Florence's Mayor Dario Nardella tweeted (in Italian) that no-one had been harmed in the incident on Lungarno Torrigiani.
He appealed to motorists to remove their cars from the nearby area, and said the water supply to part of the city centre was cut off.
Regional paper Corriere Fiorentino reports that thousands of residents are without water.
The first sign of the problem came not long after midnight, when the police were called, the mayor told Italy's Ansa news agency.
The major collapse took place at 06:14 local time, Mr Nardella said.
National League side Southport have named former Chester boss Steve Burr as their new manager.
[NEXT_CONCEPT]
A section of the embankment of the River Arno in central Florence collapsed on Wednesday morning, sending part of the road and at least 20 parked cars into a newly formed ditch.