Officers, some mounted on horses, lined up to divide a large number of people waving union flags from a small group of Yes supporters.
A number of people draped in union flags later left the area and began spreading on to the nearby streets, with many marching down St Vincent Place as police followed.
Union supporters and people sporting Yes badges argued over the referendum result and a chorus of Rule Britannia was countered by the Scottish anthem, Flower of Scotland.
Roads around the square were closed as police dealt with the incident.
A spokeswoman for Police Scotland said: “Officers continue to manage the situation and prevent further disruption. Most have now dispersed, a small group remains with police in attendance.”
Chief Correspondent Alex Thomson was at the scene: