The Senate had no authority to declare war at their own initiative, that decision was a privilege assigned to the Comitia Centuriata, one of the popular voting assemblies, and a ritual of throwing a spear across the border would follow a vote in favour of war. However, one example of when they did was Octavian using the priviliges he held to tell the Senate to do so against Cleopatra of Egypt.
So right there you have an encapsulation of the change from democratic confirmation of aggression to military directive. As Princeps Senatus, Augustus as First Senator had the ability to suggest a war should be undertaken pretty much when he wanted, and given his overwhelming presence on the Roman political scene, he was likely to be listened to. He also had the right to intervene where-ever he wanted in defence of Rome's security, which given their policy of offence being the best defence, and holding the highest right to command (imperium maia), he could assume command of Rome's military whenever he wanted. Since the Comitia Centuriata were essentially by-passed, that requirement lapsed into history, which the SEnate were perfectly happy about.
Later Imperators - the name meant 'Victorious General', not 'emperor, though did not confer any official power whatsoever) were emulating Caesar and Augustus hence followed similar prerogatives.