The incident happened at around 10.15am on the northbound A74(M) at junction 15 near Beattock.
Police Scotland said two men were taken to Wishaw General Hospital and one to Dumfries and Galloway Royal Infirmary. Their conditions are not yet known.
There are not thought to have been any passengers on board the coaches at the time of the incident.
One lane of the road, a key route into Scotland, remained closed this afternoon to allow for repairs and for debris to be removed. It is believed the vehicles have been recovered from the scene.
A force spokeswoman said: "At 10.15am police and emergency services were called to a report of a road crash involving a lorry and two coaches.
"There are three confirmed injuries."
She added: "Police will continue their inquiry into what caused the crash. If anyone has any information they can contact police on 101."