Most were for anti-social behaviour and public disorder, the force added.
Thousands of people took part in processions on Saturday which shut down city centre roads and prompted counter-protests.
Police repeated their condemnation of “outbreaks of racist and sectarian singing” by some supporting the processions and said they are investigating a number of “hateful” videos showing this.
Chief Superintendent Mark Sutherland, divisional commander for Greater Glasgow, said: “Officers in Glasgow made a total of 14 arrests during the Orange Order processions on Saturday, September 18, the offences were primarily for public disorder and acts of anti-social behaviour, but also included sectarian-related breach of the peace.
“There were outbreaks of racist and sectarian singing by some of those who attended to support the processions, this is utterly unacceptable and we completely condemn this behaviour.
“We are aware of videos circulating that show some of this hateful singing and are already investigating a number of these.
“Our main priority throughout the event was to maintain public safety and ensure minimum disruption to the wider public.”
Up to 800 police officers were deployed to manage the event, which involved marches past Catholic churches.
Members of Call It Out, a campaign group opposing anti-Irish and anti-Catholic bigotry, held vigils outside churches on the routes.
In 2018 a Catholic priest was attacked outside St Alphonsus’ Church in the city as an Orange walk marched past.
Crowds lined the streets in the city centre for the marches on Saturday including on George Street and West George Street, and there was a large police presence at Glasgow Green.