An explosion was heard at the Royal Bank of Scotland branch on Langholm High Street at about 01:40 on Friday.
Police sealed off the area while carrying out their investigations.
A short time after the raid a green Land Rover was found burning on the B6318 road about three miles south of Langholm's Skippers Bridge.
Firefighters extinguished the blaze.
Police said the amount of money taken had been "substantial" but declined to specify the exact sum.
Det Insp Scott Young said: "At this stage we are carrying out a number of enquiries into this crime and we are asking for the help of the public on a number of fronts.
"We are keen to hear from anyone who may have seen what took place at the bank at around 01:40 this morning.
"We have reports of a green coloured Landrover R445 YJC being at the scene of this crime and then driving off south shortly afterwards."
He said they were keen to track the movements of the vehicle before and after the break-in.
"We are also keen to hear from anyone in the town who may have seen any suspicious activity in or around the area of the bank over the past days or even weeks," he added.
Ch Insp Stephen Stiff, operational commander for the area, said: "This is a very rare crime for our area and totally out of character for the Langholm area.
"We have officers in the Langholm area carrying out door to door enquiries and speaking with the public to offer reassurance.
"The High Street, which forms part of the main A7 road is still closed and will remain so until we have carried out our full scenes of crime examination."
Diversions have been put in place to allow traffic to bypass Langholm.
Police said they were aware this would cause disruption but that it was necessary in order to gather evidence.
Thieves have made off with a "substantial quantity" of money after blowing open a cash machine during a bank raid in southern Scotland.