Police said staff at Uddingston's Thorniewood Road branch were left "extremely shaken" after being threatened during the raid at 14:40 on Friday.
They said the man fled with a four-figure sum.
He escaped in a waiting silver Vauxhall Vectra, which was later found burnt-out 5.6km (3.5 miles) away.
Officers discovered the car in Commonhead Road, near Easterhouse.
Det Cons Iain Hughes from Wishaw CID said: "No-one was injured during this incident, but the bank staff are extremely shaken by the robbery.
"The man responsible is described as white, around 5ft 10ins in height and of slim build with short fair hair."
Mr Hughes said officers were following a positive line of inquiry but have appealed for anyone with information about the robbery to come forward.
A man armed with a knife has robbed a branch of Royal Bank of Scotland in South Lanarkshire.